OPTIMAL SAMPLED-DATA CONTROL OF
DISTRIBUTED PARAMETER SYSTEMS

Thesis for the Degree of Ph. D.
MICHIGAN STATE UNIVERSITY
KWANG YUN LE
1971

w
w

LIBRARY

Michigan State
University

"I—‘a-l

  
      

  

This is to certify that the
thesis entitled

Optimal Sampled-Data Control of
Distributed Parameter Systems

presented by

 

Kwang Yun Lee

has been accepted towards fulfillment
of the requirements for

Ph. D. degree in Systems Science

 

 

fla/ézf 6 flat/U

Major professor

Date October 22, 1921

0-7639

 

 

 

ABSTRACT

OPTIMAL SAMPLED-DATA CONTROL OF
DISTRIBUTED PARAMETER SYSTEMS

BY

Kwang Yun Lee

This thesis is concerned with the sampled-data control
prOblem for distributed parameter systems with quadratic cost
criteria, where the system operators are the infinitesimal
generators Of semigroups Of operators.

An equivalent discrete-time problem is formulated in the
variational framework. The existence and uniqueness of an
optimal control is proved and a necessary condition for Opti-
mality is derived. The optimal control is given by a linear
feedback law Of sampled states. The feedback Operator is
shown to be the bounded, positive semi-definite and self-
adjoint solution of a nonlinear Operator difference equation
Of Riccati type. This operator is represented by an integral
operator whose kernel satisfies an integro-difference equa-
tion. These results are shown to hold for the control prOblem
on the infinite time interval with an additional assumption.

The results Obtained above for general distributed con-
trols are then specialized to the case Of pointwise control.

The Optimal discrete-time pointwise control is given by a

Kwang Yun Lee

simplified linear feedback law which depends on the control
point location. A finite dimensional eigenfunction approxi-
mation is Obtained by a suitable choice of cost functional.
The structure Of feedback controls for this approximation is
composed of an Observer which is independent Of control
point location and a gain matrix which depends on control
point location. These are illustrated by an example Of the

scalar heat equation.

OPTIMAL SAMPLED-DATA CONTROL OF
DISTRIBUTED PARAMETER SYSTEMS

BY

Kwang Yun Lee

A THESIS

Submitted to
Michigan State University
in partial fulfillment Of the requirements
for the degree Of

DOCTOR OF PHILOSOPHY

Department Of Electrical Engineering and
Systems Science

1971

ACKNOWLEDGMENTS

The author would like to express his sincere apprecia-
tion to his major professor, Dr. Robert O. Barr, for his
constant guidance and assistance during the preparation Of
this thesis and for his friendly and thoughtful help and en—
couragement throughout his years of graduate study.

Gratitude is also expressed to Dr. Shui-Nee Chow for
his unselfish interest, encouragement, and suggestions

during the period devoted to this work.

Thanks are also due to Dr. John B. Kreer who initially
brought this problem tO author's attention through his teach-
ing, to Drs. A. V. Mandrekar and P. K. Wong for providing him
a profound mathematical background through their teaching,
and to Dr. Robert A. Schlueter for his interest in this work

and helpful discussions.

The author also wishes to express his gratitude to Dr.
Herman E. Koenig for assisting him as an academic advisor
and for providing him financial support granted by Michigan

State University and the National Science Foundation.

Mrs. Glendora Milligan Of Mathematics is deserving of

special acknowledgement for her excellence in typing this thesis.

Finally, the author thanks his wife Sangwol and his son

Eddy, who provided a great deal Of help in the form of love,

patience, and encouragement.

ii

TABLE OF CONTENTS

Chapter Page
I. INTRODUCTION . . . . . . . . . . . . . . . . . . 1
II. MATHEMATICAL BACKGROUND. . . . . . . . . . . . . 7
2.1 Distribution and the Sobolev Spaces . . . . 7
2.2 Differential Operators. . . . . . . . . . . 13
2.3 State Equations . . . . . . . . . . . . . . 16
2.4 Semigroups . . . . . . . . . . . . . . . . 18
2.5 Representation Of Solutions for

Distributed Parameter Systems . . . . . . . 23
III. OPTIMAL SAMPLED-DATA CONTROL . . . . . . . . . . 27
3.1 Control Problem . . . . . . . . . . . . . . 27
3.2 Sampled-Data Formulation. . . . . . . . . . 30
3.3 Discrete-Time PrOblem (DTP) . . . . . . . . 33

3.4 Decoupling and the Riccati Operator
Difference Equation . . . . . . . . . . . . 42
3.5 Control on the Infinite Time Interval . . . 54
3.6 The Riccati Integro—Difference Equation . . 69
IV. SAMPLED-DATA POINTWISE CONTROL . . . . . . . . . 75
4.1 Pointwise Control Problem (PCP) . . . . . . 75
4.2 The Solution of PCP . . . . . . . . . . . . 78
4.3 Approximation . . . . . . . . . . . . . . . 84
4.4 An Example for PCP. . . . . . . . . . . . . 90
V. SUMMARY AND CONCLUSIONS. . . . . . . . . . . . . 96
LIST OF REFERENCES . . . . . . . . . . . . . . . . . . 99
APPWDIX . O O O C C O I O I O O I O O O O O O O O O O 104

iii

LIST OF FIGURES

Figure Page
4.2.1 Optimal Feedback Pointwise Control System . . . . 83
4.4.1 Sampled-Data Control System for Example 4.4 . . . 94

iv

CHAPTER I

INTRODUCTION

One Of the many prOblems arising in Optimal control is
the prOblem Of controlling distributed parameter systems.
In general this prOblem is concerned with determining the
system inputs which minimize some given performance criteria
when the systems are described by partial differential equa-
tions. Most Of the research in this problem has been
oriented tO the use Of continuous, rather than discrete, con-
trols in the time evolution. In many cases Of practical
interest, however, it is strongly desired to control dis-
tributed parameter systems by means Of discrete-time controls.
Furthermore, since the nature Of continuous—time evolution is
desirable and should be retained, sampled—data control, that
is, sampling the continuous-time data, is considered tO be
the most practical scheme. Examples Of such cases are the
control Of traffic flow in an urban freeway using digital
speed metering and the control Of temperature distribution in
a rolling plate employing scanning-type thermocouple measure-
ments. In both Of these examples the controls are piecewise-
constant controls, that is, controls which are constant for
each time interval. Moreover, controls are determined on
the basis Of sampling continuous data. Other examples for

which sampled-data control might be applied are systems in

2

which measurement data for feedback control is not avail-
able continuously in time, systems having a long range Of
time evolution which implies that periodic sampling is
inherent in the process, or systems employing a digital
computer in the feedback lOOp as a part Of the controller.
Biological systems such as the growth Of cereal leaf
beetles, algal production in lakes, detritus processing in
streams, or plankton distribution in nutrient pools require
experimental measurements at discrete instants in time and
thus typify processes Of the first type. POpulation models
for ecological systems Of great diversity would require
dynamic equations having a long range Of time evolution.

The purpose Of this dissertation is to formulate the
sampled-data control problem for distributed parameter
systems and to present a general approach tO Obtaining the
solution of this problem. The system to be considered is
described by an evolution equation in function space, where
the system Operator is an infinitesimal generator Of a semi—
group Of bounded linear Operators. The cost functional is
quadratic in the deviation Of the state distribution from
a desired distribution and in the control energy.

Many approaches have been used tO the solution Of
general continuous-time distributed parameter systems.
Basically these might be classified into four types Of
approaches. The first one is the approach Of Butkovskii's
[B-B], [8-9]. The distributed parameter systems he considers

are those described by nonlinear integral equations.

3

For these systems he has employed the calculus of varia-
tions to develop a maximum principle, that is, a necessary
condition for the Optimal control. Sakawa's [5-3], Yeh and
Tou's [Y-2], and Yavin and Sivan's [Y-3] works fall into
this category.

The second approach is that Of wang [W-l] and wang and
Tung [W—Z]. They derive a maximum principle for distributed
parameter systems described by partial differential equa-
tions by suing a dynamic programming procedure, which has
been extended to function spaces by Bellman and Kalaba [B—ll].
Using the same approach Kim and Erzberger [K-4] have Obtained
a set Of functional equations analogous tO the matrix Riccati
equation for lumped systems. The works Of Katz [K—S], Egorov
[E-l], [E-2], Sirazetdinov [8-4] and Brogan [B-lO] are in the
spirit Of this approach.

The third type is the well-known system theoretic con-
cepts initiated by Balakrishnan [8-4], who develOped a
general theory Of Optimal control problems in Banach spaces
using the theory Of semigroups Of linear Operators and has
applied some of the results to a class of control problems
for distributed parameter systems. Specifically, he con—
siders both the time-Optimal and final value control prOblem.
Fattorini [F-3], Axelband [A-3] and Freedman [F-2] have
worked along these same lines. Russell [R-3], Lukes and
Russell [L-4] and DatkO [D-2] have considered the quadratic
cost functional for the systems in Hilbert spaces and

derived feedback control laws.

4

The last type Of approach is the variational approach
Of Lions [L-l]. He characterizes the control problems Of
systems described by partial differential equations as
variational prOblems, and generates a new maximum principle,
giving necessary and sufficient conditions to the solution
Of the variational prOblems associated with distributed
parameter systems. Greenberg [G—l] extended the results Of
Lions tO the systems whose spatial differential Operators
are infinitesimal generators Of semigroups Of Operators.

Most Of the works that have been performed for a decade
are concerned with continuous-time systems. Recently
Matsumoto and Ito [M-l] formulated a discrete—time pointwise
control prOblem from a second order parabolic system which
has a green's function associated with it. They Obtained a
feedback control law using dynamic programming.

The lack Of works in the discrete-time Optimal control
Of distributed parameter systems has motivated this research.
We shall consider general distributed parameter systems
whose spatial differential Operators are infinitesimal gen-
erators Of semigroups of Operators. We shall formulate the
discrete-time control prOblem in the variational framework
which was adopted by Lions for continuous—time problems, and
will generate new results for the control problem associated
with discrete-time distributed parameter systems. we shall
also show that distributed systems driven by finite dimen-
sional controls (the pointwise control prOblem) fall within

the framework of this formulation and the results Obtained

5
for a general class of controls are specialized to the case
Of pointwise control in a straightforward manner.

The outline Of the thesis is as follows. Chapter II
is devoted tO the mathematical background which provides
the basic structure Of distributed systems and their trans-
formations intO ordinary differential equations in infinite
dimensional function spaces. The characterization of solu-
tions tO these equations is provided using the theory Of
semigroups Of Operators.

In Chapter III we formulate a sampled-data control
prOblem, and thus an equivalent discrete-time control prOb—
lem. The necessary condition for Optimal control is derived,
and the control is given by a feedback law in which the feed-
back Operator is the solution Of a Riccati Operator differ-
ence equation. The remainder Of the chapter contains a
discussion Of the behavior Of Optimal solutions when the
terminal time approaches infinity and the integral represen-
tation Of the feedback Operators.

The results Obtained above for general distributed con-
trols are then specialized tO the case of pointwise control
in Chapter IV and a pointwise feedback control is Obtained
which is Of simpler form than the distributed feedback
control from a computational point Of view. By a suitable
choice Of cost functional, the finite dimensional approxima-
tion by an eigenfunction expansion is Obtained. The approx-

imation is illustrated by an example Of the scalar heat

equation.

Chapter V contains a summary Of the results in the

thesis and recommendations for further research.

CHAPTER II

MATHEMATICAL BACKGROUND

This chapter is devoted tO the mathematical foundation for
the developments which will be presented in the sequel. Par-
tial differential equations are formulated into ordinary dif-
ferential equations in the function spaces in order to Obtain
the analogy Of distributed parameter systems with the lumped
parameter theory, that is, state and state space, matrix Op-
erators, state equations, transition matrices, and the variation
Of constants formulae. Section 2.1 is concerned with the
concept Of state in distributed parameter systems and the dis-
cussion Of particular spaces Of generalized functions which serve
as state spaces. The spatial differential Operators are defined
on the state spaces in Section 2.2. Parabolic partial differ-
ential equations are converted into ordinary differential equa—
tions in function spaces in Section 2.3. In Section 2.4 the
concept Of a semigroup Of Operators is introduced. In Section
2.5 the Characterization of solutions for the distributed para-

meter systems with the aid Of semigroup theory is considered.

2.1 DISTRIBUTION AND THE SOBOLEV SPACES

In finite dimensional systems the state is a point in a
finite dimensional Euclidean vector space. But the state Of a
distributed parameter system at each instant Of time is a

function defined on the given spatial region, or, in other

7

8

words, the state is a.pOint in an infinite dimensional
(function) space. Since quadratic cost criteria are Of
interest, this space is chosen to be the Hilbert space of
square integrable functions on the spatial region Of
definition. Again, as will be shown, this space is not
quite suitable for distributed parameter systems, but
certain subspaces, namely the Sobolev spaces, are suitable.
Let D be an Open set in Rn with boundary 8D.
Throughout it is assumed that D is a bounded, Open set
‘with boundary 3D which is a C00 -manifold Of dimension

(n-l). The symbol 2 = (z zn) denotes the spatial

1. .
variable in D. Further let C3(D) be the space Of in—
finitely differentiable functions Of compact support on D,
i.e., m E C3(D) vanishes on the outside Of the compact
support Of [>(cf. [Y-l],p.62). The space Of bounded linear
functionals on C3(D), i.e., the dual Of C;(D), is called
the space Of distributions, or generalized functions, on D,
and is denoted by BTD). A element F E 3(D) has the form

F(cp) = f 13(2) cp(z)dz v m e cgm)
D

where f(-) is some Lebesgue integrable function on D.
We present two Of the prOperties on the space Of
distributions. First, the space Of square integrable
functions on D, L2(D), is a subset of PTD). This is
easily seen by noting the fact that C3(D) C L2(D) (i.e.,
any infinitely differentiable function with compact support

on D is square integrable on D) and, therefore, the dual

9

space Of L2(D) must be contained in the dual space Of
C8(D), namely fi(D). Since L2(D) is its own dual the

following inclusion relation holds
(2.1.1) cgm) c L2(D) c 19(9) .

The second prOperty is the differentiation Of distribution.
If F E 3(D), the distributional derivative or the general-
ized derivative Of F (with respect to zi, i = l,2,°-°, n)

is defined by (cf. [Y-l], p. 49)

 

O _ Oco on
(2.1.2) —Oz. F(cp) — - F(Oz.) v co 6 com).
1 1
Remark 2.1.1: The above notion is an extension Of the

 

usual notion Of the derivative. For, if the function f is

continuously differentiable with respect to 21, then we

have
Oz f cp f Oz Oz 1 n
l l D 1
(2.1.3)

O
—-f(z).cp(z)dz ---dz =F (.9) .
I J;)Oz1 l n Of/Ozl

as may be seen by integration by parts Observing that m(z)
vanishes identically outside a compact subset Of D. Thus,
in VieW’Of (2.1.1) we may define the differentiation (in the
sense Of distribution) for all elements Of L2(D).

This generalized approach to differentiation can be
extended tO any order Of differential Operators:

Corollary 2.1.2 [Y-1,p.50]: A distribution F E.3(D)

is infinitely differentiable in the sense of distributions

10
defined above and

(2.1.4) Dq Pup) = (-1) ‘9’" qucp) v cp e cgw) .
where
n
(2.1-5) q = (q10q20°'°t (In): Iql = § qio
1—1
and
q q q
q _ l 2 ... n _ O
(2.1.6) D — D1 D2 Dn , Di — OET°

1
We now have the following definition.

Definition 2.1.3: The Sobolev space of order m,

 

denoted by Hm(D), is defined by
H‘“(D) = (F: F GL2(D), DqF EL2(D) v q, |q| gm] .

The space Hm(D) can be shown to be complete in the

topology induced by the inner product

(2.1.7) <F,G> Z <qu, qu> 2 .

Hm(D) _ IqISm L (D)
and, hence, it is a Hilbert space (cf.[Y-l], p.55). Thus,
with the completeness Of Hm(D), the Sobolev spaces can be
considered as the state spaces Of distributed parameter
systems.

Next we describe the subspaces Of Hm(D) which incor-
porate with certain boundary conditions. For each x E Hm(D)
we may associate the traef of x on OD as well as that Of

its normal derivative 'j;k x, for 1 g_k g_m - 1, and in
On

this way characterize the image Of Hm(D) by the map

11

-l

Ox Om x

(2'1'8) X " x'OD' On|OD' ' anm-l OD'

where g%- is the outward normal derivative on OD. This

characterization requires Sobolev spaces of non—integral

order and it is therefore essential to introduce such spaces.
The SObOlev spaces Of non-integral order are defined

by means Of Fourier transformation. We first consider the

space Hm(D) ‘with D = Rn. The Fourier transform Of x,

32(9) is defined by

(2.1.9) :x<g) = f exp(-2wj<g-z))x(z)dz.

n

R
where (5-2) = £121 + --- + ann, the usual inner product
on Rp.. It can be shown that

(2.1.10) .7qu(§) = (21rj) IqI Qq Jx(€) V x E L2(Rn) .

q ql qn
where C = Q1 --- Qn . This results in an alternative

definition Of the Sobolev space, namely
Idu9)={xzdfixm)e19mh quIgm}

or, equivalently,

(2.1.11) em(an) = {x: (1 + |g|2)’“/2 :x(g) e L2(Rn)}.

Note that there is nO restriction in allowing m tO be an
integer or positive in (2.1.11), rather than requiring it to
be any number in R. Thus, the SObOlev spaces can be defined

for any number m €.R.. In general Hm(D) is defined by

(2.1.12) Hm(D) = (x: (1 + |g|2)‘“‘/2 Jx(€) €1.20)” ,

12

which is a Hilbert space with inner product

2 m/2 2 m/2
(x.y) = <(1+|c|) JX(C).(1+|C|) :y(g)> 2 .
Hm(D) L (D)

It can be shown that the dual space Of Hm(D) is H-m(D),

i.e., (Hm(n))’ = H‘m(n) (cf.[Y-l,p.99, p.155], and [L-3]).
We are now in a position to state the trace theorem:
Theorem 2.1.4 [Lions-Magenes, L-2]: For any

x E Hm(D), we may define in a unique manner its traces

X _x . . . a x
)OD' On OD' ' ann-l OD'

 

Moreover, we have

Bk _ -1
%' EH 2(OD),ng_<_m-l,
On OD
ka
and the map x 4 f—fiE,. O g_k g_m—l] is a linear, continuous
On
m—l m_k_l
mapping Of Hm(D) onto H H 2 (OD).
k=0

Lions and Magenes [L-Z] also showed that the kernel Of

the map (2.1.8) (i.e., the space Of x E Hm(D) such that
'94E = O, O g_k g_m—1)- is the closure of Cm(D) in the
Onk an O

norm of Hm(D). We denote this subspace by H$(D). Thus

k
(2.1.13) Hg(n) = {x eH‘“(D):‘-5-—LC

k =O,nggm-1}.
On

Ia.
Since H3(D) is a closed subspace Of Hm(D), and therefore
a Hilbert space (with the inner product Of Hm(D)), it may
just as easily be considered as a candidate for a state space

as Hm(D). Here, the condition imposed on the boundary OD

13

in the definition Of Hm(D), i.e., ———-
0

On OD

is called the Dirichlet boundary condition.

I)
.0
O
l/\
x
/\
a
l
f”

2.2 DIFFERENTIAL OPERATORS

In this section we will discuss the prOperties Of
spatial differential Operators, which play the role in dis-
tributed parameter systems which matrices play in lumped
parameter systems. It has been shown in the preceding
section that a differential Operator Of order m is every-
where defined and closed on Hm(D). However, the Operator is
not bounded, which gives rise a major distinction to matrices
in finite dimensional systems.

Let aq(z) be a real valued function, where q is the
n-tuple defined by (2.1.5). Define the formal differential
Operator A of order m:

(2.2.1) A = Z a (z) Dq,
IslSm q

where Dq is the differential Operator defined in (2.1.6)

and

Z = Z + Z +---+ >3
lqlsm C? ‘1

1qu lg 1 Iq|=m

Similarly we define the formal adjoint differential Operator
Of A (cf.[C-l], p.59), denoted by A', as
(2.2.2) A' = Z} (_1))QI Dq a (2).

Is I‘m q

In general, the formal adjoint Operator A' is not equal to

l4

* *
the adjoint Operator A , where A is defined by

*
<x.Ay> = <A X.y>

Hm(D) Hm(D).

It can be shown, by means Of Green's formula (cf.[C-l,p.63],

and [Y-l,p.50]). that

<x.Ay> = <A’x.y> + C.
Hm(D)

where the constant C depends on conditions at the boundary
OD.. In the case Of Dirichlet boundary conditions, discussed
in Section 2.1, C = O and thus A' = A .

If aq(z) G Lm(D), i.e., if it is essentially bounded

th

(cf.[R-l], p.112), then the m order differential Operator

(2.2.1) is said to be elliptic (cf.[D—l], p.1704) if
2) a (z)Cq # o v C e Rn, z e D.) c # O.
Iq=mq

Note that this is a condition on the highest order term, i.e.,
the terms containing partial derivatives Of order m.. If
we restrict our attention to elliptic differential Operators
which contain only even order partial derivatives, we define
the concept Of coercivity in the following manner:

Definition 232.1: If A is an elliptic differential
Operator of the form
= Z) a (z) D(:1

|q|52p q

where aq(z) = 0 if qu # 2k,. for k = O,l,--o, p, then

A

A is said to be coercive if

(2.2.3) (-1)k 23 aq(z)gqg - a Z gq

for some a > 0,. for k = O,l,°°°, p, and for all
C E Rn and z E D.

This concept Of coercivity describes the property Of
Operators more commonly referred to as negative definiteness,

namely the condition

<Ax.x> < - a M2

Hm(D) _ Hm(D)
for some a > O and for all x E Hm(D). It might be noted
that just as negative definiteness Of a matrix implies that
the eigenvalues Of the matrix lie on the negative real axis,
the spectrum of a coercive operator is a subset Of the left
half—plane. We also define the strong ellipticity which has
milder condition.
.Qefinition 2.2;2: If A is an elliptic differential

equation Of even order 2p, then A is said to be strongly

elliptic if

(2.2.4) (-1)p Z) aq(z)gqg-a 23 gq
Isl=2p

for some a > 0,, and for all C E Rn and z 6 D.. Note
that (2.2.4) is a special case Of (2.2.3), which applies only
fOr the highest order terms. Thus, coercivity implies strong
ellipticity, but the converse is not true.

A more general Operator than those considered above

which will be introduced later is the differential Operator

16

which plays the role Of an infinitesimal generator Of a
semigroup Of Operators. This will be studied in Section

2.4 and 2.5.

2.3 STATE EQUATIONS

Utilizing the concepts Of state, state space, and
system differential Operators discussed in the preceding
sections, we describe the state equations in the form Of
partial differential equations with an additional time
variable.

Let x(t) be a function defined on t E [O,T] with
values in the SObOlev space Hm(D), i.e.,
x(t) E Hm(D) V t 6 [O,T]. For each t E [O,T], x(t) may
be considered as a point in the function space Hm(D). We
define the space L2(O,T; Hm(D)):

Definition 2.3.1: The space Of square integrable

SObOlev space-valued functions is

T
L2(O,T;Hm(D))=[x:x(t) e Hm(D) v t e [O,T], J” ”x(tHlZ dt<oo}.
O Hm(D)

Note that this is a Hilbert space with inner product

T
= ,J‘ <x(t).y(t)> dt.

<x.y>
L2(0.T;Hm(D)) o H’“(D)

In order tO describe the dynamics (i.e., evolution in
time) Of distributed parameter systems, we may again introduce
the notion Of distributions on [O,T] as we did on D in
Section 1. If we consider the space Of infinitely differen-

tiable SObOlev space-valued functions with compact support in

17

[O,T] and its corresponding dual space Of distributions,
which may be denoted by fi[0,T], then
L2(O,T:Hm(D)) C fi[0,T] and the following SObOlev space Of
SObOlev space—valued functions may be defined (cf.[L-l],
p. 102).

Definition 2.3.2: The Sobolev space of Sobolev space—
valued functions on [O,T], denoted by ‘W(O,T), is

defined by

W(O,T) = (x: x e L2(O,T:Hm(D)), x e L2(O,T:Hm(D))}.

11.
dt
This is a Hilbert space with inner product

dxéx

<X'3’>w(o.'1') = <X'Y> 2 < O?’ dt>L2

L (o.T:H“‘(D)) (0.T:H‘“(D))
we are now in a position tO describe partial differen-
tial equations by ordinary differential equations in the space

Of SObOlev space-valued functions. The parabolic equations

are Of the form:

(2.3.1) fiiépze) = A x(t,z) + f(t,z)

where A is an elliptic partial differential Operator in
the spatial variable 2. If x(t,z), t 6 [O,T], z E D is
assumed to be the element x(t) € W(O,T), then (2.3.1) has
the equivalent formulation as the ordinary differential

equation in L2(O,T: Hm(n))

(2.3.2) %x(t) = A x(t) + f(t)

where f 6 L2(O,T: L2(D)). If the initial condition is given

18
by x(O,z) = xo(z) E Hm(D), then we write the initial

condition Of (2.3.2) as x(0) = x0.

2.4 SEMIGROUPS

In this section we shall introduce the notion of
semigroups which play the similar role Of the transition
matrices in finite dimensional systems. Let I be a Banach
space and let 6(1) be the Banach algebra Of endomorphisms
Of I, i.e., the space Of bounded linear transformations on
I to itself (cf.[H-l], p.51).

Definition 2.4.1 ([B-l], p.7): If a mapping

§(t): [O,m) * 6(1) satisfies the following conditions:
(2.4.1) (i) @(t1 + t2) = §(tl)§(t2), t1, t2 Z_O,
(2.4.2) (ii) §(O) = I, I = identity Operator,

then {§(t), t 2.0} is called a one-parameter semigroup Of

Operators in 6(1). The semigroup {§(t), t 2_O} is said

tO be Of class (CO) if it satisfies the further property
(2.4.3) (iii) s-lim {>(t)x = x , x E I,
t40+

refered tO as the strong continuity Of §(t) at the origin.
In the sequel we shall generally assume, unless other-
wise stated, that the family Of bounded linear Operators
{§(t), t 2_O} mapping I into itself is a semigroup Of
class (C0), thus that all three conditions Of the above

definition are satisfied. We further state some of the

properties Of the semigroup in the following:

19
Lemma 2.4.2 ([B-l], PrOposition 1.1.2): (a) Hi(t)H is
bounded on every finite subinterval Of [O,w).
(b) For each x E I, the vector-valued function
9(t)x on [O,m) is strongly continuous.

(c) One has

Ill

(2.4.4) w

o inf %-log He(t)n = lim % log Hs(t)u < m.

t>O t-ooo

(d) For each m > w

0' there exists a constant Mm

such that for all t 2_O

(2.4.5) Hut) H g MUD e‘”t .

In part (b) Of the lemma we have seen that the Operator
function §(t) is continuous on [O,m) in the strong

Operator topology, i.e., lim U§(t)x - §(t0)xH = O for any
t4t

O
to 2_O and for all x E I (cf.[B-l],p.290). Thus the family
{2(t), t 2_O} is Often called a strongly continuous semi-
group in 6(I). If, in addition, the map t 4 §(t) is

continuous on [O,m) in the uniform Operator topology, i.e.,

lim IIMt) - Mto)“ = O for any to >_o (cf.[B-l], p.290),
bit

0
where the norm is the usual induced Operator norm on I
(cf.[T-l], p.86), then {§(t), t 2_O} is said to be a
uniformly continuous semigroup in 6(I). In case the norms
Of the semigroup Operators are bounded uniformly with respect
tO t, i.e., H§(tH|g_M (M a constant larger than or equal
to one) for all t 2_O,, then [§(t), t 2_O] is called an

equi-bounded semigroup Of class (C in 6(I), and if

0)

the constant M is equal tO or less than one a contraction

20

semigroup Of class (CO) in 6(I).

 

Definition 2.4.3 ([B-l], p.9): The infinitesimal

 

generator A Of the semigroup [2(t), t 2.0} is defined

by

(2.4.6) AXES-lim A x, A =l[t(n) -1]
110+ n n n

whenever the limit exists; the domain of .A, in symbol
DO(A), being the set Of elements x for which this limit
exists.

Lemma 2.4.4 ([B-l], PrOposition 1.1.4): (a) DO(A)
is a linear manifold in I and A is a linear Operator.

(b) If x E DO(A), then Mt) x E DO(A) for each

t 2_O and
(2.4.7) Ed; Mt) x = A il>(t)x = §(t)A x, t 20;
furthermore,
t
(2.4.8) e(t)x-x=j‘ {>(T)Axd'r, tZO.
O

(c) DO(A) is dense in I, i.e., DO(A) = I, and
A is a closed Operator.
Remark 2.4.5: (a) [B-l, p.13] If B is any Operator

in 6(I), then the Operator function

(2.4.9) Mt) =exp(tB) 21+ 55 _(.E§.Lk_, Ost<a
k=1 'k!

defines a semigroup in 6(I). Indeed, by the definition

and the properties Of the ordinary exponential function

21

one Obtains that ”@(t)n g_exp(t HBH) for all t 2_0 and
that {6(t), t 2_0} satisfies the semigroup properties
(2.4.1) and (2.4.2). Moreover, it can be shown that §(t)
is continuous, even continuously differentiable in [O,m)
in the uniform Operator tOpology. Thus {4(t), t 2.0}
forms a uniformly continuous semigroup in 6(I), its

infinitesimal generator is given by B and

.39; exp(tB) = B exp(tB) = exp(tB) B , 0 g t < 0°-

(b) [H-l, Theorem 9.4.2] On the other hand, it may be
shown that every uniformly continuous semigroup in 6(I) is
Of the form (2.4.9).

As noted in the above remark every uniformly continuous
semigroup has a bounded infinitesimal generator (cf.[D-l],
p.621), and therefore has an exponential representation
(2.4.9). But in the case the infinitesimal generator A Of
a strongly continuous semigroup is in general not bounded,
i.e., is not defined over the whole space, so that the
exponential expression (2.4.9) involving A would be meaning—
less. TO avoid this difficulty it might be valuable tO use
a limiting argument employing An which is a bounded Operator
as defined in (2.4.6). In fact it has been shown (cf.[B-l,
p.19], [R-2,p.401]) that a limiting exponential solution does
exist, which is referred tO as Hille's first exponential

formula:

(2.4.10) §(t)x=lim exp(tA)x Vx EI, tZO,
n40 n

22

where the convergence is uniform with respect to t in any
finite interval [O,b]. SO every strongly convergent semi-
group has the representation (2.4.10).

An alternate representation for the semigroup is in
terms Of the resolvent R(A;A) Of the infinitesimal gener-
ator A. The resolvent is the Operator R(1;A) = (AI—A)-1
defined for all values Of A for which the inverse exists

(cf.[H-l], p.123). It can be shown (cf.[H-l], p.342) that

the resolvent Operator is the Laplace transform Of the semi—

group:
R(1:A)x = I e_Xt §(t)x dt, x E I
O
for all Re(l) > w , ‘where w is defined in (2.4.4),

0

moreover, we have the Laplace inversion formula

ij

i (t)x = s—lim a“:
Y*“

2wj w-jy R(1,A)x d1
for each x E DO(A) and t > 0 with w > max(0,w0).

One of the important questions is under what conditions
will a closed linear Operator A be the infinitesimal

generator Of a semigroup Of class (C The Hille-Yosida

O)°
theorem ([H-l], p.364) tells us that a necessary and sufficient
condition fOr a closed linear Operator A to generate a semi-
group {§(t),t 2.0} Of class (C0) is that there exist real

numbers M and m such that for every real 1 > w, and

(2.4.11) ]]R(1:A)nH SM (A-m)’n , n = 1,2,...

23

we now can determine whether the spatial differential
Operators Of Section 2 are infinitesimal generators Of
semigroups. Dunford and Schwartz [D-l, p.1767] showed that
the necessary condition for the elliptic partial differential
Operator A defined in (2.2.1) to be an infinitesimal gener-
ator is
(2.4.12) (—1)m/2 Z a (z)§q g 0 , z E D, Q 6 Rn .

Isl=m q

Note that the condition for strong ellipticity (2.2.4)
clearly satisfies the necessary condition (2.4.12). In this
respect a series Of extensive works has been devoted to the
strongly elliptic partial differential Operators (cf.[A-l],
[A—2], [B-2] and [B-3]). An important result is that if
{§(t),t 2_0] is the semigroup of Operators generated by a
strongly elliptic Operator, than the bounded Operator §(t)

has the exponential bound
(2.4.13) Hut) H g M e’it

where M. and 1 are positive constants (cf.[F-l], p.72
and p.158).
It remains to characterize the solutions Of partial

differential equations with the aid Of semigroup theory.

2.5 REPRESENTATION OF SOLUTIONS FOR DISTRIBUTED
PARAMETER SYSTEMS
In this section we will characterize solutions of

partial differential equations. First let us consider the

24

homogeneous equation Of (2.3.2)
(2.5.1) {<(t) = A x(t).

Then as we noticed in Lemma 2.4.4 we may invoke the theory
Of semigroups Of Operators: Indeed, Phillips [P-l] has
shown that a necessary and sufficient condition for (2.5.1)
tO have a unique solution in [O,w) for each initial value
x(O) E DO(A) such that
(2.5.2) s-lim x(t) = x(O)

t-ao+
is that A be the infinitesimal generator Of a strongly‘
continuous semigroup {i(t),t 2.0] Of class (CO) (cf.

Definition 2.4.1). The solution itself is given by
(2.5.3) x(t) = §(t) x(O),

where Of course, from Definition 2.4.1 and Lemma 2.4.4,

4(tl+t2) = §(tl) §(t2), t1, t2 2_0
4(0) =1

and
d

E? §(t)x = Ai>(t)x = 6(t)Ax. x E DO(A)-

Thus the Operator §(t) is the Obvious analog of the transi-
tion matrix in finite dimensional systems. This result can be
extended to characterize solutions Of forced equations in a

form analogous to the variation Of constants formula in finite

dimensional systems.

25

we introduce the concept of measurability and Bochner
integrability for our own purpose.

Definition 2.5.1 ([H-l], p.72): A function on [O,w)
tO I is strongly measurable if there exists a sequence Of
countably-valued functions converging almost everywhere in
[O,m) to f(t).

nginition 2.5.2 ([H-l], p.78): (a) A countably—
valued function f(t) on [O,w) to I is integrable

(Bochner) iff “f(t)“ is integrable (Lebesgue). We define
(B) fat) an = Z fk “(1k 0 r)
I k=1
where f(t) = fk on Ik.’ I C [O,w) and u is a Lebesgue
measure.

(b) A function f(t) on [O,w) to I is integrable
(Bochner) if there exists a sequence Of countably—valued
function {fn(t)] converging almost everywhere to f(t) and
such that

11m] “f(t) — fn(t)]l an = o ,
n-W [0'm)
and we define
(B) f f(t) an = lim (B) J‘ fn(t) an.
I 11-900 I
Now we are in a position to discuss the solution Of a

forced system

Q
(2.5.4) int) = A x(t) + f(t), x(0) = x

dt 0 °

26

The construction Of a solution for this system can be found
elsewhere (cf.[T—Z], [K-l], [Y-l], [P-2], [F—l], and [B—4]).
We state here one Of the results due tO Balakrishnan.

Theorem 2.5.3 (Balakrishnan, 1965): Let A be the
infinitesima1.generator Of a strongly continuous semigroup
{§(t),t 2_0]. Let f(t) be strongly measurable and Bochner
integrable in every finite interval in [O,m). Further let
f(t) E DO(A) for almost every t. Then, (2.5.4) has a
unique solution given by

1:

(2.5.5) x(t) = Mt) x(O) +J‘ e(t-T) f(T) (3T.
0

CHAPTER III

OPTIMAL SAMPLED-DATA CONTROL

The purpose Of this chapter is to formulate the sampled-
data control problem for a distributed parameter system and
to solve the equivalent discrete-time problem (DTP). The
continuous-time system constrained by piecewise constant con-
trols is transformed into DTP, and the DTP is treated in the
framework Of a variational problem, i.e., that Of characteriz-
ing extremals to a given functional, constrained by an infi-
nite dimensional difference equation.

The sampled-data problem and it's equivalent DTP are
formulated in Sections 3.1 and 3.2. In Section 3.3 the
existence and uniqueness Of solutions for DTP is proved and
the necessary condition for Optimality is derived. In Section
3.4 the Optimal control is given by a feedback Operator which
satisfies an Operator difference equation Of Riccati type. The
control on the infinite time interval is investigated in
Section 3.5. In Section 3.6 it is shown that the Riccati Op-

erator equation is equivalent tO an integrO-difference equation.

3.1 CONTROL PROBLEM

In this section we will discuss the space Of controls,
and then define the quadratic cost criteria for general
distributed parameter systems. In Chapter II we have seen

that the SObOlev spaces and the elliptic partial differential

27

28

Operators are specific examples Of an abstract Banach space
I and the infinitesimal generator A Of strongly continu-
ous semigroup on I, respectively. Since we already have
an expression for the solution Of evolution equation in an
abstract Space by (2.5.5), we may begin with the general
distributed parameter systems rather than the specific cases.

Let H and U be Hilbert spaces, and x(t) E H and
u(t) E‘U be the state and control Of the system at time
t E [O,T], respectively. We denote £KX:Y) to be the space
Of bounded linear transformations from X into 'Y, and let
B(t) E £(U:H) for all t E [O,T]. We assume that A is a
closed linear Operator defined on a dense domain DO(A) E,H
and generates a strongly continuous semigroup §(t) for
tZO.

We now consider a control system

(3.1.1) fix”) =A x(t) + B(t) u(t), x(0) = x0 EH.

we further assume that B(t) u(t) is strongly measurable and
integrable in the sense Of Bochner (cf. Section 2.5). Then
by Theorem 2.5.3 the existence and the uniqueness Of solution

tO (3.1.1) is guaranteed and its solution is given by

t
(3.1.2) x(t) = i(t)xO +j i(t—T)B(¢)u(¢) 6T.
0

Next, the quadratic cost criteria weighting the state and
the control will be introduced. The notations <-,-> and
"-H ‘will be used for the inner product and the norm on H,

respectively (or on U’, which can be distinguished in

29
the context).
Definition 3.1.1: Let R 6 £(H;H). (a) [P-3,p.203]

*
The adjoint Operator R Of R is defined by

*
< Rx,y > = < x,R y > V x E H.

'k
(b) R is called self-adjoint if R = R

(c) R is called positive definite if <x,Rx> 2 aHxH
for some a > 0, V x 61H.
(d) R is called positive semi-definite if <x,Rx> 2 O,
V x E H.
If we denote the desired state distribution as xd(t) E;H,
t E [O,T], we may state the cost criterion for the system
(3.1.1) as:

T
(3.1.3) J = I [<x(t)-xd(t).o(t)(x(t)-xd(t))>+<u(t).R(t)ut)>]dt
O

+ <X(T) ‘ Xd(T)o F(X(T) " Xd(T))>o

where, for each t E [O,T], Q(t), F E £(H,H) are bounded
self-adjoint positive semi-definite Operators and

R(t) E {KU,U) is a bounded self-adjoint positive definite
Operator.

The minimization Of cost functional (3.1.3) over all
control u(t) 6.0 has appeared elsewhere (cf.[L-l], [L-4],
fD-Z], [G-l], [6-2], and [F-2]). In many cases Of practical
interest, however, it is actually desired to control distrib-

uted system by means Of discrete-time controls. Thus sampled-

data control is desired and is formulated in the next section.

30

3 . 2 SAMPLED-DATA FORMULATION

The term sampled-data is used to describe systems in
which the sampling Operation occurs between the plant and
the controller, such as systems that have a telemeter link
in the feedback loop or use a single instrument tO monitor
several variables in a sequential manner. An indirect way
Of introducing sampling is to define an index set
0 = {0,1,2,---, N} and the corresponding time set on O,
i.e., {ti} = [ti = i6: i E G], where 6 is a sampling
period. If the terminal time is finite, then. tN = T = N6,
where it has been assumed that T is an integral multiple
Of the sampling period. This assumption is not essential,
but is made for convenience (cf. [L-5]). The control
discretization on 0 requires that the inputs be piecewise—
constant functions Of time and that changes Of values Of u(t)

occur only at the sampling instants ti' that is

(3.2.1) u(t) = u(ti) E ui for t 6 [ti'

t1+1) '
Note that u(t) is strongly measurable and integrable in the
sense Of Bochner (cf. Section 2.5), hence by Theorem 2.5.3
we have a unique solution in the form Of (3.1.2). Now we
define a basic sampled-data control prOblem (BP):

Basic Problem (BP): Given a system (3.1.1) with a cost
functional J by (3.1.3), find a sequence Of controls

* * , .
u = {ui E‘U, 1 6 C} such that for all u = {ui 6'0, 1 6 o}

J(u*) = inf J(u).
u

31

It should be noted that the purpose Of using the cost
functional in continuous form (3.1.3) is to penalize the
system for error or excessive control inputs continuously
in time rather than at the sampling instants, thus to
achieve a better performance (cf.[L-6]).

This BP cannot be solved directly because the admis—
sible controls are constrained tO be piecewise-constant.
As was done in the finite-dimensional case (cf.[L-5]), we
transform this problem from a constrained one to an un-
constrained one by integrating the differential equation and
the cost functional and thus going from a continuous-thme
prOblem to a discrete-time one. The transformation is accom-

plished through the use Of solution (3.1.2) evaluated for

t 6 [ti'ti+l)' We therefore Obtain
(3.2.2) x(t) = Q(t—ti) x(ti) + D(t,ti) ui,
where for each t 6 [ti'ti+1) D(t,ti) 6 £(U;H) such that

t
(3.2.3) D(t,ti) ui =j‘ Mt-w) B(T) ui d'T’.

t.

1
Letting t = ti+l.' we Obtain a state difference equation
(3.2.4) xi+1 = 2 xi + Di ui, x0 6 H
where xi = x(ti), Q = 4(6) = i(ti+1-ti), and Di = D(ti+1,ti).

If u(t) is constant over the sampling period, using (3.2.2)
we Obtain the following expression for the cost criterion

(3.1.3):

32

N—l
(3.2.5) J = 1:30 [<X1'Qixi> + 2 <Xi'Miui> + <ui’Riui>

_ 2 <xi,sixdi> - 2 <ui.sixdi> + <xdi.Eixdi>]

+ <xN.F xN> - 2 <XN.F XdN> + <XdN.F xd > .

N
where for each i 6 0,. Qi'si’Ei 6 £(H;H), Ri 6 £1U;U),

Mi 6 £(U:H), and P1 6 £(H:U) such that

ti-i-l :1-
(3.2.6) Q.x. I e (t-ti)Q(t) Q(t-ti) xi dt,

1 1
t.
1
t1+1 *
(3.2.7) Mini = fl: 2 (t-ti)Q(t)D(t,ti) ui dt,
1
ti+1 ' *
(3.2.8) Riui = j [R(t) + D (t,ti)Q(t)D(t,ti)]ui dt,
ti
1+1 *
(3.2.9) Sixd. = f e (t—ti)Q(t) xd. dt,
1 t, 1
1
ti+1 *
(3.2.10) Pixd. = f D (t,ti)Q(t) xd. dt,
1 t. 1
l
t1+1
(3.2.11) Eixd. = j Q(t) xd. dt.
1 t. 1
1

Note that for each i 6 0 Q1.- and E1 are self-adjoint
positive semi—definite Operators, and R1 is a self-adjoint
positive definite Operator. Thus the continuous—time system

(3.1.1), with the cost functional (3.1.3) and the control

constraint (3.2.1), has been transformed into the discrete-

time system (3.2.4) with the cost functional (3.2.5).

33

We therefore define a discrete—time problem (DTP) which is
equivalent to BP.

Discrete-Time PrOblem (DTP): Given a system (3.2.4)
with a cost functional J by (3.2.5), find a sequence of
controls u* = (u: 6‘0, 1 6 0} such that for all
u = {ui 6HU, i 6 o]

J(u*) = inf J(u) .
u

Throughout this dissertation we will be concerned with
solving DTP .

3.3 DISCRETE-TIME PROBLEM (DTP)

In order tO solve DTP we will introduce the function
spaces on which DTP can be handled easily. We denote x
and u to be sequences Of states and controls on 0,
respectively, such that x = {x0,x1,---, xN-l] and
u = {u0,u1, o-o, uN_1} with xi 6 H and ui 6.U . Let
£2(0,N:H) be the family Of all functions x on O with
values in H.

Remark 3.3.1 [D-l,p.257]: L2(O,N;H) is a Hilbert
space with usual addition and scalar multiplication, and with

an inner product defined by, for x, y 6 £2(0,N:H),
N-l
(3.3.1) <x.y>£2(o'N;H) = 1:30 <xi.yi>H .
Similarly, we may define a Hilbert space £2(0,N:U) with
inner product analogous tO (3.3.1).

TO prove the existence and the uniqueness Of solution

for DTP 'we require that the solution Of difference equation

34

(3.2.4) depends continuously on the control. Hence we
prove the following lemmas.

Lemma 3.3.2: The mapping u 4 x Of 22(0,N;U)
into £2(O,N;H) defined by the difference equation (3.2.4)
is continuous.

Proof: The solution x Of (3.2.4) can be expressed as

i—l
(3.3.2) xi = §(1)xO + k2: 2(1—l—k) Dk uk, x0 6 H,
=0
where Q(i) E Q(ib). Let xl,x2 6 22(0,N;H) be the solutions

(3.3.2) corresponding to controls u1,u2 6 £2(0,N;U), respec—

tively. Then we deduce that

N-l
”x1 - x21122 = 23 11x; _ xi”;
2 (O.N:H) i=0
N-l i-l
= Z Z Q(i-l-k) (ul-uz) HZ
i=OHk=0 Dk k k '
Ni1[iZ-31 H 1 2) H 2
g Q(i-l—k) ( - ]
i=0 k=0 Dk uk uk

Since 4(1) and Di are bounded we have
H§(i-l-k)Dk(ui-ufi)fl g H§(i-l-k)DkH Hui-ufi” .

Note that ”4(1)“ is bounded by Lemma 2.4.2, and HDkH is
bounded by the uniform boundedness theorem (cf.[Y—l], p.69),
i.e., H§(i)H S.M and HDkH g_d . Thus we have

i-l
Z) HQ(i-l-k)DkH2 g,i M2d2 < a
i=0

so that by Schwartz' inequality

35
42

- i-l ' ( )Dk(u1 L112: ]
2: Q ' _1_k _ )

( ) 1231) 12151111 212
i i Bk] i A k kl
. 2 2 1 2 2
g_1 M d Hu -u H 2 .

Substituting (3.3.5) into (3.3.4), we have

2 N-l
Hxl-xz]! 2 g 23 i Mzdzllul-uzflzz
‘ 2 (O,N;H) i=0 2 (O,N;U)

2 2 2

g_N M d Hul

.2 2
'2 (0.N:U)

or equivalently,

llxl-XZH 2 g c Hul-UZH

(OIN7U)

which implies that the mapping u 4 x Of £2(0,N:U) into

22(0,N:H) is continuous.
Q.E.D.

Lemma 3.3.3: The mapping u 4 xN Of 22(0,N:U) into
H defined by the difference equation (3.2.4) is continuous.
Proof: Let xi, x; 6 H be the terminal state due tO

controls ul, u2 6 12(0,N:U), respectively. Then by (3.3.2)

N-l 2
1 2 1 2
I - = II '23 @(N-l-k) (u -u ) I
IXN XNHH .k D]. k k l,
N-l 2
g [ 23 l!{>(N-1-k)Dk(u]1(-ufi) n]

k=0

N-l 2
[23 IIuN-i—kwku Hul—uzll] .
k=0 ' ' k k

|/\

36

Using the same argument as in the proof Of Lemma 3.3.2

we have

[.1

1 2 ‘2‘ 1 2
- H g.N Md u -u .
HXN XN'H H ӣ2(O.N:U)

Lions [L—l,pp.6—10] proved a general existence and
uniqueness theorem for controls minimizing a certain cost
functional. He also showed that this theorem covers the
existence and uniqueness Of Optimal controls for the con-
tinuous—time control problem. Now, the discrete—time
prOblem (DTP), defined in Section 3.2, will be shown to
fall into Lion's framework in the function spaces
£2(O,N:H) and 12(O,N;U).

Theorem 3.3.4: The discrete—time problem (DTP),
defined in Section 3.2, has a unique solution u* 6 £2(0,N;U).
TO prove this we need the following definition and

lemmas due to Lions. Let V be a Hilbert space.

‘nginition 3.3.5: A continuous symmetric coercive bi-
linear form w(u,v) is a continuous function in both argu-
ments which maps V x V into the reels for which there

exists a C > 0 such that

w(u,u) 2_C HuH2 V u 6‘V

w(u,v) = v(v,u) V u, v 6‘V
If we consider a functional

(3.3.6) C(u) = W(u,u) — 2L(u), u 6'V,

37

where L is a bounded linear functional defined on ‘V, then
‘we have the following:

Lemma 3.3.6 (Lions, [L—1]): If v(u,v) is a continuous
symmetric coercive bilinear form, then there exists a unique

*
u 6‘V such that

C(u*) = inf C(u) .
u€V

Lemma 3.3.7 (Lions, [L-l]): If the hypotheses Of Lemma
*
3.3.6 are satisfied, then the minimizing element u 6'V is

characterized by
*
(3.3.7) 1r(u ,v) = L(v) V v 6V .

The proofs Of above lemmas will be given in Appendix for
reference. Now we return to the proof of the theorem.

ggoof Of Theorem 3.3.4: We may write the cost functional
(3.2.5) with inner products in the function spaces 22(0,N;H)

and 12(O,N:U), i.e.,

J = <x,Qx> 2 + 2<x,Mu> 2 + <u,Ru> 2
L (OoN:H) fl (OoN:H) fl (0.N:U)
(3.3.8) —2<x,Sx > -2<u,Px > +<x ,Ex >
d 22(0.N:H) d £2(0.N:U) d d 12(0.N:H)
+ <xN,FxN>H -2<XN,deN>H +<de,deN>H ,

where 9.5.3 6 2(42(0.N:H):22(0.N:H)).R e 4(42(0.N:u):12(o.N:U)).
M e £(LZ(O,N:U):L2(O,N;H)), and p e £(£2(0,N;H):£2(0,N:U))

such that for i 6 {0,l,°°°, N-l}

Qx {Qixi}' MU = {Mini}: Ru = {Riui} I

Sxd [Sixdi], de = {Pixdi}, Exd = {Eixd.}'

l

38

Note that Q and E are self-adjoint positive semi-definite
Operators, and R is a self—adjoint positive definite
Operator. we will simplify the notations by deleting sub-
scripts in the expression Of norms in the spaces 22(0,N;H)
and £2(0,N:U) unless they are necessary.

Let xu denote the response Of the system (3.2.4) due
to control u 6 22(0,N;U). We define the bilinear form

1r(u,v) on £2(0,N:U) x 22(0,N;U) to be

1r(u,v) E (xv-XO,Q(xu-x0)> + (xv—xO,Mu> + <xu-xO,Mv>
(3.3.9) + «cg—x3 , F(x§-xg)> + <v.Ru> .
and a linear functional L(v) on £2(0,N;U) to be

(3.3.10) L(v)

III

- <xV-XO.QXO> - <x9.Mv> - <x§-x§. Fx§>
v 0 v 0
+ (x -x ,Sxd> + <xN-xN,deN> + <v,de>.
Then the cost functional (3.3.8) becomes

J(u) = W(u,u) - 2L(u)

(3.3.11) + <x°,ox°> + <x§,Fx§> - 2 <xO,Sxd>
O
— 2 < ,Fx > + (x ,Ex > + (x ,Fx > .
"N a“ d d dN a“

Since the last six terms are independent of 'u, the prOblem
is equivalent tO minimizing J1(u) = w(u,u) — 2 L(u). The
continuities Of v(u,v) and L(v) follow from Lemma 3.3.2
and Lemma 3.3.3. Clearly w(u,v) is symmetric. The coer-

civity Of W(u,u) follows from (3.3.9) and the definition Of

39
R(t), i.e.,

Mum) =J‘ [<xu (t) -x°(t) .Q (t) (x“(t) -x°(t))> + <u (t) .R(t)u(t)>]dt
O

+ <x“('r) — x°(T). F<x“(cr) - x°(T)>

T T 2
2,] <u(t).R(t)u(t) > at 2_c I Hu(t)n dt

0 O

N-l 2
= c E Hui)! = c Hull 2 .
i=0 fl (0,N:U)

Since w(u,v) satisfies the hypotheses Of Lemma 3.3.6, there

*
exists a unique u 6 22(0,N;U) such that

*
J (u) = inf J (u). Q.E.D.
l 2 1
u6£ (0,N:U)
Next we derive the necessary condition for Optimality
which is analogous to the result for finite-dimensional

systems.

*
Theorem 3.3.8: If u e 12(O,N;U) is the Optimal

control fOr the discrete-time problem (DTP) with Optimal
*
response x 6 12(0,N:H), then there necessarily exists a

*
unique adjoint state p 6 22(0,N;H) such that

* _ -1 _ * * -1 * *
* * * *
pi ‘ i pi+1 + Qixi ' Sixdi + Mi ui'

(3.3.13) p;

F(}C; " de) o

_ * 'k 'k
where R 1 is the inverse Of JR, and Mi' D. and i

are the adjoints Of Mi" Di and 9, respectively.

40

'1:
Proof: We note x satisfies

'3?

3314 * - * * -
( . . ) xi+1 — Q xi + Diui , xO — xO .

From the proof Of Theorem 3.3.4, H(u,v), defined by
(3.3.9), is a continuous symmetric coercive bilinear form.

Hence, by Lemma 3.3.7, the Optimal control must satisfy
* 2
v(u ,v) = L(v) V v 6 i (0,N:U),
or equivalently,
* * * 2
(3.3.15) v(u ,v—u ) = L(v-u ) V v 6 L (0,N:U).
Further, let us introduce the adjoint equation

*
Pi = Q pi+1 + Qixi ‘ Sixdi + Mi ui '

(3.3.16) pN = F(xN - de) .

Here pi is called the adjoint state and it should be noted
that a unique solution p 6 12(0,N:H) exists for (3.3.16).
In fact, by changing i to N-i and realizing 9* is a semi-
group (cf. [B-l], p.47), we can have an explicit solution for
pi for all i 6 O, 'which is a similar form to (3.3.2). Let
us denote (xu,pu) as the solution pair of a system (3.3.14)
and (3.3.16) due to a control u 6 12(0,N:U). Forming the

v u

inner product on 12(O,N;H) between pu and x - x we

Obtain

41

N-l u v u
<pu,xV-xu> = Z <pi’Xi—X1>H
i=0
-Ni1<§*u xV-xu>+<Mu Xv_u>
‘ i=0 pi+l' i i H ' x
(3.3.17) + < qu - Sxd, xV - xu>
“231 u v u v u
= i=0 < pi”. “Xi—Xi» + < Mu.x -x >

V

+<qu-Sx x-xu>.

d I

Note that the left—hand side Of (3.3.17) can be expressed as:

N-l

u v u __ u v u u v
<P 'X 'X > ‘ 5’0 <pi+l'xi+l ‘Xi+1>H ‘ <PN'XN" >H °
(3.3.18)
N-l u v u u v u
= Z) <pi+l'xi+1 ‘Xi+1>H ' <F(XN ‘XdN)'xn"xN>
i=0

Equating (3.3.17) and (3.3.18), using (3.3.14), and letting
*
u = u we Obtain

N_1 * * *

*
23 <p‘i’+1,ni(vi-ui)>H = <qu -Sxd, xv—xu >
i=0

(33.19)

* *

'k
+ <F(x:-de), XII-XE > + < Mu* , xV-xu > .
Now from (3.3.15),

* * *
1T(u,v-u) -L(v—u)
u* v u* u* v u*
=<Qx -Sxd. x -x >+ <F(xN-de).xN-XN >

(3.3.20)
* *
+ <Mu* ,xv—xu > + (x11 , M(v-u*)>

* * *
+<Ru ,v-u >-<de,v-u>=0.

42

Combining (3.3.19) and (3.3.20), we Obtain
N-l * *
u * u *

* * *
+ <Ru ,v-u > - <de, v—u > =0,

or equivalently,

NE} * u* * * u* *
. [(Dipi+1' Vi"“i>H + <MiX ' Vi"ui>H
1=0
(3.3021)
* * *

d.'
1

Since (3.3.21) hold for all v e 22(O,N:U), we obtain

* 'k

*_ -*u-*u
(3.3.22) Riu. — Pix Mix Dipi+l .

Moreover, since R is positive definite (cf.Definition
3.1.1) it has an inverse (cf.[Y-l],p.43) and so (3.3.22)

reduces to (3.3.12). Q.E.D.

3.4 DECOUPLING AND THE RICCATI OPERATOR DIFFERENCE EQUATION
In this section we derive a feedback form Of the

Optimal control given by (3.3.12). The feedback Operator is

shown tO be bounded, self-adjoint and positive semi-definite,

and the cost functional is expressed in terms of the feedback

Operator.

We define bounded Operators on H:

1

(3.4.1) 8.

_ *
é - DIR. Mo '
1 1 1 1.

1

(3.4.2) T. Q. - MiRi

M*
1 1 i'

43

Lemma 3.4.1: The Operator Pi 6 £(H:H) is bounded
self-adjoint and positive semi-definite for all i 6 0.

Ppppf: The self—adjointness and boundedness comes
directly from those Of Qi and Ri . It is clear that
for all i 6 o

t.
1+1
l.i =f [<x(t).Q(t)x(t)> + <u(t).R(t)u(t)>]dt 2 o
t.
1

or equivalently,
L.1 = <Xi'QiXi> + 2<Xi'Miui> + <ui,Riui> 2_O

for all u. 6;U . Now let u. be given by u. = —R71fo. .
1 1 1 1 1 1

Then we have

L = < x (Q - M R'1M*)x > 2 0
' i' i i i i i

. . . 2
<x1. I‘lxl> O.

which implies that Pi is positive semi-definite. Q.E.D.
Using (3.4.1) and (3.4.2), the system Of equations

(3.3.14) and (3.3.13)can be simplified into the form:

_ -1 * -1
xi+1 ‘ ®ixi ' DiRi DiPi+1 T DiRi Pixdi
(3.4.3)
- 8* r '1
pi ‘ iPi+l + ixi ‘ Sixdi T MiRi Pixdi
x3 = h 6 H, pN = F(xN-de): 1 6 {3, 5+1, -°-, N}, s 6 O.

This system admits a unique solution pair
(x,p) 6 £2(s,N;H) x 22(s,N:H). This fact is easily seen

if the cost functional J in (3.2.5) is defined on the

44

interval [s,N] instead Of on [O,N]. The system (3.4.3)
has the following prOperties.

Lemma 3.4.2: The mapping h 4 (x,p), solution Of
(3.4.3), is continuous from H into 22(s,N:H) x 22(s,N;H).

Proof: 'Without loss Of generality we let x = O .

d
Let us denote by xn(v) the state Of the system given by

(3.4.4) P x. + D.v.,

1+1 1 1 1

X
ll

xS = hn, on [s,N].

For a fixed v, if hn 4'h,
n . 2
(3.4.5) x (v) 4 x(v) 1n .6 (s,N;H).

Let J:(v) denote the cost with control v and initial
condition h at time s . Let un and u be the optimal

h
control for an(v) and J2’, respectively. Then

h h h h
anm") = inf an(v) g an(u) and JS In(n) -» J};(u) from

(3.4.5). Hence
-——- h n h
(3.4.6) lim an(u ) g JS(u) = inf J:(v) .

But
hn n NE} n 2
JS (u ) 2_C i=5 Hui“ ,

which when combined with (3.4.6) shows that un belongs to
a bounded subset Of £2(s,N;U) as hn 41h. Then we can

choose (cf.[Y-l], p.126) a subsequence uk such that

(3.4.7) uk -o w weakly in 22(s,N;U).

45

Therefore, xk(uk) 4 x(w) weakly in 12(s,N;H) and hence
h
lim J km“) > Jh(w) .
-—- s - s
which when combined with (3.4.6) shows J:(w) g_J:(u) and

hence necessarily w = u. Therefore,

un 4 u weakly in L2(s,N:U),
hn n
(3.4.8) Js (u) 4 J}S‘(u) ,

and

xn(un) 4 x(u) weakly in £2(s,N;H),

pn(un) 4 p(u) weakly in £2(s,N;H).

This proves the continuity Of the linear mapping h 4 (x,p)
from H into £2(s,N:H) x 12(s,N:H). Furthermore, (3.4.8)
implies that un 4'u strongly in 12(s,N:U) and hence the
mapping h 4 (x,p) is in fact continuous in strong
topologies. Q.E.D.
Corollary 3.4.3: For h 6 H, let (x,p) be the

solution of (3.4.3). Then the mapping
(3.4.9) 'h 4 PS

is continuous from H into H .

gpppf: The proof follows from the fact that the
mapping (4.7) is the composition of the mapping h 4 (x,p)
and the mapping (x,p) 41pS . But
(x,p) 6 12(s,N:H) x £2(s,N;H) implies that for every

1 6 [s,N] ”xi" and “Pi” are bounded. Hence we may take

46

subsequences x?,.p2 such that

x
-:3

4‘;i weakly in H V i 6 [s,N],

P

14:: H

45.1 weakly in H V i 6 [s,N] .
The second equation Of (3.4.3) becomes in the limit,

_ *.._ _
pi = ®i p1+1 +Ti xi on [s,N],

hence, we may take 5: = p for all i 6 [s,N], and in
particular 5; = pS . Thus the mapping (x,p) 4pS is
continuous from £2(s,N;H) x 22(s,N;H) into H.
Q.E.D.

Now we have the feedback representation Of the Optimal
control.

Theorem 3.4.4: The Optimal control u* 6 £2(0,N:U)
for the discrete-time problem (DTP), defined in Section

3.2, is given by the feedback form

—1 —l l —1
ui = -[R; M: +Ri D.1K.H+1(I+D R. D. :Ki+l) 8i]x.l
-l * -l * 11—1
-1-1 -l*-1 -l
+[Ri Pi -R. D:Ki+1(l+ D. R. D. Ki“) DiRi Pi]xd ,

i

where for i 6 0,. Ki is the solution Of the Riccati—type

Operator difference equation

* -1*
(3.4.11) K. = D. i+1(I + D. 1R.1 D5

-1
1 1 @. + T.,

K1+1) 1 1

KN”:

47

and 9.1 is the solution of the linear difference equation

_ * * -1
g.-[®i- @iK i+1(I+D.Rl D:iK+

l) ”lDi Ri1D*1]g'

-1

(3..412) +[®:iK +iil(1+DR D:K. 1 ‘1 ‘1

1+1) D1R1 P1+M1R1 Pi_si]xd '

gN = — F}:

Proof: From the continuity of h 4‘ps,. pS can be
written uniquely (cf.[L-l], p.135) in the form

(3.4.13) p = Ksh + gS .

S

where Ks 6 £(H;H) and 9S 6 H.. Since 5 is arbitrary in

o and h is the evaluation of X5, (3.4.13) implies that
(3.4.14) pi = Kixi + gi V 1 6 o,

where (x,p) is the solution pair of the system (3.4.3).

Using (3.4.14) we can rewrite the system (3.4.3) as

x. = ®.x.+-D.R71P.x -—D.1R.11D*i (K
1 1 1 1

1+1 1 d. )'
i

i+lX i+l +gi+1

+M.R71P.x
1 1 1 di

) + F.x.-—S.x

*
(3'4'15) Pi ®1(K1'1+1"'1+1+‘3’1+1 1 1 1 di

. . + .
lel 91'

PN KNXN + 9N = Fxbq - F§flHq: x0 6 H.
Rearranging the first equation in (3.4.15), we obtain

—1*

=(I+D. R. D. K. 1 '1 "*1

[®ixi+DiRi P.x —D. 1Ri D. 19

(3.4.16) x 1 di

1+1 1+1) i+l]'

where the inverse is well-defined since Ki+l is positive

48
semi-definite by the next theorem (Theorem 3.4.5). Substi—

tuting (3.4.16) into the second equation in (3.4.15) and

rearranging terms, we have

* -1 —1
[Ki - ®iK Kii+l(I+D Ri D. *iKi+1) ®i - I‘i]x.l
(3417) =-g+@*g -@*K (1+Di'i'*RlDi.K )_i;leRlD
° ° 1 1 1+1 '1 1+11+1191+1
* -1* -1 -1 -1
+ ®iK i+i1(I+D Ri D. 1Ki+l) DiRi Pixdi-Sixdi+MiRi Pixdi

Since xi is arbitrary in the sense that it depends on an

arbitrary choice of x0,

satisfies (3.4.11) and (3.4.12), respectively. The feedback

it is necessary that Ki and gi

control (3.4.10) follows from (3.3.12), (3.4.14), (3.4.11) and

(3.4.12). Q.E.D.

Next we examine the properties of the feedback operator
KS and express the optimal cost in terms of Ks
Theorem 3.4.5: The feedback operator Ks on H is

self-adjoint, positive semi-definite and bounded.

Proof: (Self-adjointness). Let (xl,p1) and (x2,p2)

be solution pairs corresponding to initial conditions h1 and
hz, respectively, for system (3.4.3) with xd = O . Then
N-l *
0= '2 (pi-(@ip]: —rixi,xi>
i=s 1+1
N-l N-l N- l
=Z<p1,x>-Z<pll,ox>—23<I‘llx.2>.
._ 1 1 1
1-s 1: 3 i=5
N- 1
_ 1-1 * 2 l
_ (psncs 2>-- <PN XN>- 1:21 <pi+l’Di Ri Dip —.z:3 (I‘ixi, x: >,
:5 i+l> i_s

Hence by (3.4.13),

49

N-l N-l

2 _ 1 2 1 -1 1
.h >‘<FXN'XN>+£S<P1+1'D1R1

0* 2 >+ <1“ 2>
1P1+1 i=s 1x1'x1 °

(3.4.18) <Ksh1

Thus self-adjointness of Ks comes from those of F’Ri' and
T..
1
(Positivity). Let u E 22(s,N;U) be the opthmal control
with cost J:(u) of the system (3.4.3) with xd = O . Opti—

mality implies that u satisfies the necessary condition

it *
(3.4.19) Riui + Mixi + Dipi+1 = O .
If h1 = h2 =‘h, the equality (3.4.18) becomes
N-l _1 *
(3.4.20) (Ksh,h> = @131 ,xN> + 1E5 <pi+1,DiRi Dipi+l>
N-l N-l *
+ .Z) <Qixi’xi>'-.Z) <MiRilMiXi'xi>°
1=s 1=s
But, by virtue of (3.4.19), we deduce
2 <Mu,x> 2 + <Ru,u> 2
L (SIN7H) 1% (SIN7U)
(3.4.21)
N-l N-l
_ -1 * -1 *
" " .23 <M1R1 M1"1"‘1> + § <131+1'I’1R1 I’11’1+1>°
l—S l—S
Combining (3.4.20) and (3.4.21), we obtain
£ (s,N;H) £ (s,N;H)
(3.4.22) + <Ru,u> + (F 1 >
£2(SIN:U) XN )CN H
= J‘Qm) 2 o .

which proves the positive semi—definite property of KS .

50

(Boundedness). Since, by Lemma 3.4.2, the transforma-

tions h 4 x, h 41p, and ‘h 4 xN are continuous in the

strong topology, we have ”x” 2 g_C1HhUH
L (s ,N;H)
Hp” g_c ”h” , and H H g,c HhH . Since F,
2(S'N;H) 2 H XN H 3 H

R, and F are bounded, we have
<rx,x> s MlCi HhUZ, <p,DR'1D*p> s Mzcg HhHZ, and
<F15q,xN> s M3C§ uh“2 . This implies that, by (3.4.20)

<K.h h> s (MC2 + M c2 + M c 2) Hfhll2

s ' 1 1 2 2 3 3

Thus K is bounded. Q.E.D.

3

Theorem 3.4.6: The Optimal cost of system (3.4.3) with

 

initial state h at time s is given by
*
Jim) = <KSh.h> + 2 <95 . h> + cps

where KS and gS are solutions of (3.4.11) and (3.4.12),

respectively, and ms is the solution of

_ -1—1-1 \
mi — wi+1 ' <pixdi'Ri PiX di>+b<Dii R P1Xdi'HiDiRi P ixdi/
- 2< D R. 11D. H. D. R. 1P. lx >+-2< D R“1D*x >
g1+1'11 1 1 1 di 91+1' 1 1 1 d.
1
-1-1—1
(3 4° 23) + <9 i+1DiRi DiHiDiRi Digi+l> <g1+1'Di Ri D*1gi+1>'

¢N=<FXdNI XdN):

where Hi E £(H,H) is a self-adjoint positive semi-definite

_-1-1
operator such that H. Ki+1(I + D. JR D* iKi+1)

Proof: From the second equation of (3.4.3) we have

51

-1
<I‘x-Sx+MR Px,x-x>
d d d £2(s,N;H)
(3.4.24)
N-l
= . <p1 ‘ ® 1P1+1' X1 ‘ Xd.>H
1=s l

The left hand side of (3.4.24) can be expanded to

- * -1
<Qx,x> — <MR lM x,x> - <Sxd,x> + <MR de,x>

-1 * -1
- <Qx,xd> + <MR M x,xd> + <Sxd,xd> — <MR de,xd>.

Using the first equation of (3.4.3), and using the similar way
for the derivation of (3.4.18), the right hand side of

(3.4.24) can be shown to be
<Kh+g.m»M$( -x ). -x >~GW -X 1x >
s s XN dN xN dN XN dN dN

-1 * -1
- <Q x,xd>+-<MR M x,xd>-<MR de,xd>4-<Sxd,xd>

N- 1 _1 N-1_1
+ Z3 <P1+1'D1R1 Pixd.> ‘ Z) <P1+1'D1R1 D P1+1>
i=5 1 i=3

Thus (3.4.24) yields

<QX.X> - 2<Sxd.x> + <F(xN - de).XN - x N

= - <Sxd.x> + <MR-1M*X.X> - <MR-1de,x>

(3.4.25) > + (Ksh + gS,h>

- (F (XN - de) onN
N-1_1 N-1_1
+ Z) <p D. 1R1 P. x > - Z3 <pi +1.D1 R. D. .p1+1>

I
1_ 5 1+1 i d1 i-s

From the necessary condition (3.3.12) it can be shown that

2<x,Mu> + <u,Ru> - 2<u,Px >

d

._ * .-
(3.4.26) = - <MR 1M x,x> - (de,R 1de>
_1 N— l
+ 2<de,R M* x> + Z3<<p
1=s

D. R. “1D1p.

i+1' i+1.>

52
Thus, the cost functional (3.3.8) becomes, by (3.4.25) and
(3.4.26),

*
J2<u ) = <KSh.h> + <gs.h> - <F(xN-de).de>

(3.4.27)

-1 -1
(de,R de> + <MR PXdIX>

l

N—l
- <Sxd,x> + Z) <p1+l'DiRi P1Xd > .

1=s 1
When we use (3.4.12), (3.4.14), (3.4.15), and (3.4.16), then

(3.4.27) can be expressed as

*
J:(u ) = <Ksh.h> + 2<gs.h> + <deN, xd >

N
+ gE}[-<P x R‘lp x > + <D R‘lp x H D R'lp x >
._ i d.' i i d. i i i d.' i i i i d.
1-5 1 1 1 1
(3 4 28) - 2<g. D. 1R11D* H. D. R. 1P. 1xd >
° ' i+1' i i i 1 d1
+ 2<g D1 R11D* x >+<g D. 1R11D*1H. 1D.1R11D*1g >
i+1' 1 d1 i+1' 1+1
-< Di R11D* >].
gi+1'1gi+1
where H = (I + D. 1R11D* 1K. )_1 hence we rove (3 4 23)
i Ki+1 1+1 ' p ' '

Q.E.D.
Note that H1 is bounded, self-adjoint, and positive semi—
definite because of K1, F1 and the Riccati equation (3.4.11),

i.e.,

illi

<H1®ixi'®iyi>'

and

53

*
(xi,®.H.®.y.>

<xi'(Ki-ri)yi> 1 1 1 1

which implies H1 is bounded, self-adjoint and positive
semi-definite by those properties of Ki and Ti.
Remark 3.4.7: (a) The equation (3.4.13) is reduced to

p = Ksh when x

s d = O, and to pS = gS when h = 0.
(b) The existences and uniquenesses of the solutions
of the Riccati equation (3.4.11) and the linear equation
(3.4.12) are the consequences of the strong continuity of the
transformation h 4‘ps.
With the Riccati Operator equation on hand, we will
consider one way of solving the Operator equation. If we
assume H is a separable Hilbert space, then there exists a

basis {mi]i:l’ mi 6 H.. such that any element x e H has

a unique representation

co
x = Z) x.m.,

j=1 33

where xj = <x,mj> (cf.[R-l], p.212). Thus, we may consider
an element x E H to be alternatively represented by a
infinite dimensional vector g_ with jth component Xj'

If L is a any linear Operator on H.. we have for x EIH,

Lx = L Z3x.m. = Z3 x.Lm..
j=1 J 3 j=1 J 3

Now, Lmj is an element of H so that

54

where L'j = < Lm.,mi >. Thus L}: may be represented by
1

3
LEE. where L. is the infinite matrix with ijth element
Lij . Similarly if U is a separable Hilbert space with

on
a

basis {wi}1=1' each element u E‘U may be considered to
be an infinite dimensional vector ‘EJ and the control
Operator D may be considered to be the infinite matrix ‘2
with ijth element Dij = < ijnyi >.

Let us, for the purpose of illustration assume that
DTP is time invariant and R = I. Then we may rewrite the

Riccati operator equation (3.4.11) as the infinite dimensional

matrix Riccati equation

K —®*K I+DD*K '1
—i‘——i+ (— ———'

where all of the matrices are uniquely determined in the
fashion prescribed above.

It is possible to truncate above matrices and solve the
resulting finite dimensional matrix equation for an approxi-
mate value of the 5i matrix. An alternative way of solving

the Riccati equation will be considered in Section 3.6.

3.5 CONTROL ON THE INFINITE TIME INTERVAL

In this section we will develop a treatment of the
discrete-time problem (DTP), defined in Section 3.2, on the
infinite time interval, i.e., on 0 = 000 = {0,1,2,---]. we

., P., and E.

assume that the Operators Di' 01' Mi' Ri' S1 1 1

are uniformly bounded on O, and that F = O .

55

Let £2(O,w;H) be the family of all sequences
}

x = [x ‘with xi 6 H such that Z3uxiuz < m.
o . w

i i€o
Remark 3.5.1 [D-l,p.257]: £2(O,w:H) is a Hilbert
space with the usual addition and scalar multiplication, and

with an inner product defined by, for x, y € £2(O,m;H),
(3.5.1) < x.y >£2(O,°°;H) = g < xi.yi >H.
Similarly 22(O,w:U) is a Hilbert space with inner
product analogous to (3.5.1).
We make the following definition.
Definition 3.5.2: (a) A control u defined on O is
said to be admissible if u E L2(O,w:U).
(b) A state x is said to be a solution of system
(3.2.4) if it satisfies (3.2.4) with an admissible control
u and initial condition xo E H,, and if x(u) 6 £2(O,m;H),
where x(u) = [x(u)i]ieo.
Notations: The state of the system will be denoted by
N N

x (to emphasize the dependence on N), that is, x is

the solution of (3.2.4) on = {O,l,2,°°', N}. The cost

ON

functional in (3.2.5) is denoted by JN(u). Let (xN,pN)

be the unique solution pair of the system (3.4.3) on UN

and let X? and g? be the corresponding Operator and

function Ki and gi in (3.4.11) and (3.4.12) respectively.
On the infinite time interval we make the following

hypothesis, which was trivially guaranteed on the finite time

interval.

S6

Hypothesis I: For every u E £2(O,m;U) and x0 E H,
there exists a unique solution of system (3.2.4) on 0.

It should be pointed out that stronger hypotheses have
been used in the continuous—time prOblem (cf.[L-4] and
[D-ZJ).

Lemma 3.5.3: Hypothesis I implies that the mapping

 

u 4 x defined by the difference equation (3.2.4) is
continuous from £2(O,m:U) into 12(O,m;H).

Egggf: Let T be the mapping u 4 x(u). Clearly T
is linear. It remains to show that T is bounded. Define
Tn(u) = {x1(u), x2(u),---, xn(u)} E 22(0,m:H). Clearly Tn
is linear. Assuming x0 = O and by a simple calculation we
can show that Tn is bounded for all n, i.e.,
"Tum”! g CnHuH for all u e :2 (0,°°;U) and for all n,
where Cn is a constant which depends on n (cf. Lemma
3.3.2). Since for each u e 12(O,°°:U) u'rnm)” is bounded by
a constant ”x(u)” for all n“, by the uniform boundedness
theorem (cf.[R-l],p.l96) there is a constant C such that
”Tn“ g_c for all n . Again by Yosida [Y—l, p.69, Corollary
2] T is the strong limit of the sequence {Tn} and T is a
bounded linear operator. Q.E.D.

Remark 3.5.4: Lemma 3.5.3 implies that x E £2(O,m:H)

 

and hence Jm(u) < m for all u E £2(O,m:U). The requirement
Jw(u) < m is the hypothesis adopted by Lukes and Russell
[L-4] and Datko [D-2].

NOW‘We have the existence and uniqueness of DTP.

57

Theorem 3.5.5: Assume that Hypothesis I holds. Then
the discrete-time prOblem (DTP), defined in Section 3.2,
on the infinite time interval has a unique Optimal control
u00 E £2(O,w:U)-

m: Because of Lemma 3.5.3, Lemma 3.3.6 can be used
and the proof is identical to that of Theorem 3.3.4.

we now give a sufficient condition for Hypothesis I to
hold.

Theorem 3.5.5: If ”Q” < 1, then Hypothesis I will
hold true.

Proof: We are concerned with showing the existence and

uniqueness of solution x(u) E 22(O,w:H) satisfying

(3.5.2) Xi+1 = @xi + Diui: x0 E H .

(Existence). Let HQ“ = a < l . Let x(O)N E £2(O,N:H)
be the solution of (3.2.4) with zero control and
x(u)N E 12(O,N;H) be the solution of (3.2.4) corresponding to
a control u E L2(0,m;U) and x0 = O . Then the solution of

(3.2.4) can be written in the form
(3.5.3) x(u)N = x(O)N + Q(u)N.

N .
Now for x(O) , we derive

N-l N—1

_ N N N N
0 — £2; <x(°)i+1'x(0)i+1> - SEE <¢x(o)i,,x(0)i+l>
N-1 N-1
N 2 N N
2- 1:30 ”x(O) 1+1” " iEO ”éx (0)1” ”x(O) 1+1”

(con't.on next page)

58

(con't. from previous page)
N-1N-1 1/2
2 Z IIx(0)1f‘II2 - IIx II2 - a(. N21 IIx(O)NII 2 Z IIxIOIITT “2)

N-1
2 Z.“ IIx(O)I:II2 - HxOII2 — a N21 IIXIofi:
i=0

i=0
N-l
_ N 2 2
- (1-a)i§ollx(0)iH - IIXOII .
hence,
(3.5.4) IIx(O)N|I 2 g c , (independent of N).
1. (O,N;H)
Similarly for x(u)N with x0 = O we derive
N—l
i§:<x(u)N.+1,x(u)1:+l>- i§0<§x(u)N,x(u)i+l>=1:0<Diiu .qu)i+1>.
or
N231 IIx(u)i+ II2 — N21 II§x(u). NIIIIx(u) N+1II 3 N21 IID. lu. lIIIIx(u)i+ 1II ,
i=0 i=0 1:0
or
1/2
”2 N-l 2
l\IZlII:c(u)I:+H21\I§31I|}<(u)NiZ> II§(U)I:+1II >
i=0 i=0 i=0
N-l 1/2
3 12:30IIDJ11L ..u II2 :21 =0le (u “)i+1II2 ,
or
N-l N-l N-l 1/2
12 Him”? 1”2..1 '2: IIX(U)I:+1II2 in=0 2 IID u. II 2) x
i=0

1 2
(:ZIIM()+1II2 I/.

59

or
N-l N—l m
2 A N 2 n 2 2
(1 - a) iiioIIxm) 1+1“ 3 iizoIIDiuiII g iEOIIDiuiII .
Hence
N-l co
(3.5.5) 23 II;‘E(u)1;LTII2 3 c1 >3 IIuiIIZ g c.
i=0 i=0

Therefore, from (3.5.3), (3.5.4) and (3.5.5), we have

(3.5.6) IIX(u)NII 2 _<_ CZIIuIIZ 2 g c.
2 (o.N:H) 2 (0.00m)

Now if

AN
x.
1

extension of x(u)? by O for i > N,

3N ui 1n [00N)o
l O for i 2_n,

we have

m_~u «N N. 00
(3.5.7) xi+1 — éxi + Diui - @XN5(1-N) on [0. I:

where 6(0) = l, and 6(r) = O for r # 0,. and from

(3.5.6)

(3.5.8) IGJNII2 g c.

22(0. com)
We may then take a subsequence Nn 4 m such that

N
§'n 4 x weakly in £2(O,m:H).

we then pass to the limit in (3.5.7) and hence

x = Qx. + D.u. on [O,w).
1 1 1

i+l

60

(Uniqueness). Suppose x1 and x2 are solutions of

(3.2.4) on U with control u E 22(O,w:U). Then

1 2 l 2
(3.5.9) “Xi+l - Xi+1H.S aIIxi - xi”.
' 1 _ 2 U I
Since xo - Xo" by iterating (3.5.9) we conclude that
l _ 2 .

Thus we have shown that Hypothesis I holds.
Q.E.D.

Remark 3.5.6: The assumption that HQ“ < 1 can be
well-satisfied if @(t) is the semigroup of operators
generated by a strongly elliptic operator. Because the

Operator @(t) has an exponential bound (2.4.13):

Hut)" gMe‘xt . 14.1 > o.

and hence by a suitable choice of sampling interval t = 6,
we can always have H§H = H§(6)H < 1.
Theorem 3.5.7: Suppose ”Q” < 1, then the adjoint

state p is defined in a unique manner by

*
(3.5.10) pi = Q pi+1 + Qix.1 — Sixdi + Mini ,
and
(3.5.11) p e 12(o,oo;H).

Proof: Let us simplify the notation by defining
f. = Q.x. - S.x + M.u. 6 £2(0,w;H). Then we are concerned
1 1 1 1 di 1 1
'with showing the existence and uniqueness of p 6 £2(O,m;H)

satisfying

61

*
(3.5.12) pi = Q pi+l + fi

we note that p E £2(O,m;H) implies that p00 = 0 .
(Uniqueness). Let ”Q” = a < 1 . Suppose (3.5.12)
holds with f = 0. Then

(I) G) *
O = Z) <p.,p.> - 2} (Q p. ,p.>
i=0 1 1 i=0 1+1 1

Z iEOIIPiII - 12:30“ piflll IIpiH
w w m 1/2
Zle-IIZ - (23 1*p. "ZEN-12)
2 i=0 1' i=0” 1+1 i=0 P1"
m co co 1/2
.>_ ZIIpIIZ- (BI. II2 ZII .II2)
i=0 1" a 1:0!le i=o'pl'
2 '2 IIpiII2 - a 23 IIpiIIZ
i=0 i=0
(3.5.13) = (1 - a) Zlupiuz ,
i=0

hence p = O .

(Existence). Let qN be the solution in £2(O,N;H) of

(3 5 14) qN = §*qN + f in [o N)
'° 1 1+1 '1 ' '

where q3 = O . Then

N—l N—l * N-1
(3.5.15) 23 <qbi'.<fi’> - 2 <1 q§+l.q§> = 23 <fi.q”i‘>.
i=0 i=0 i=0

With similar arguments for (3.5.13), we deduce, from (3.5.15),

N-l N-l N-1 1/2
<1-a> ,ZIIqfiiIIZw EllfilIz ZIIqfifIIZI .
1=O i=0 i=0

or,

62
N—l N-l w
(l-a)2 Z IIqIEII2 g 23 IIfiIIZ g 23 IIfiIlz.
i=0 i=0 i=0
Hence

N’1 N 2 °° 2
(3.5.16) Z)Hqu g_c1 Z)HfiH g_c2, (independent of N).
i=0“ i=0

Now if

~N

qi

extension of g? by O for i > N,

'EN _ fi 1n [O,N),
i

O for i 2_N,
we have
“N *xN
(3.5.17) qi = Q qi+1 + E}: on [09”) I

and hence from (3.5.16),
°° .11
(3.5.18) 23 IIquI2 3 c2
i=0
We may then take a subsequence Nn 4 m such that

N
a'n 4 p weakly in £2(O,m:H).

we then pass to the limit in (3.5.17), and hence,

*
pi = Q pi+l + fi on [O,m). Q.E.D.

Now we state the necessary condition for the Optimal
control on the infinite interval.
Theorem 3.5.8: Let HQ” < l . If u00 6 £2(O,m:U) is

the optimal control for the discrete-time problem (DTP),

63

defined in Section 3.2, with Optimal response x00 6 22(0,m:H),
then there necessarily exists a unique adjoint state

p°° e £2(0,°°;H) such that

DO _ —1 — * —1 * co
co * co co m
(3.5.20) pi = Q pi+l + Qixi - Sixdil + Mini

where the Optimal response x00 E 22(0,w;H) satisfies

(3.5.21) xi+l = @xi + Diui; x0 = x0 .

Proof: Proof is identical to that of Theorem 3.3.8
hence omitted here.
Theorem 3.5.9: Let ”Q“ < 1 . The Optimal control
on

u 6 £2(O,m;U) for the discrete-time problem (DTP) on the

infinite interval is given by

no _ -1-1—1 co
ui - —[R;1*Mi + R. D. lzx +1(I + D. R. D:K Ki+l) @i]x.1
_1* -l -1 * co -1
(3.5.22) -[Ri D. - R. D:Ki+l(I + DiRi DiKi+l) D. R'i' 1*!) l]gi+l
-1171 ;1-1 -1

where for all i E O, .K: is the bounded, positive semi-

definite, self-adjoint Operator, satisfying

oo_ * -1 -1
(3.5.23) Ki—®. Ki+1II+DiR'i' D:°.°K ) (1. +1“. ,

i+l 1 1

and g: 6 £2(0,w:H) is the solution Of

64

m * *K -1
g. = [@. - @fi ;l(1 + D'1R1D*i

(3.5.24)

-1 -l —1 -l
+[C)*iKi+1(I + D. lRi D. *iKi+1) DiRi Pi + MiRi Pi"si]xdi°

Moreover, the Optimal cost on the interval [s,w) is given

by

coco (D a: co on
(3.5.25) J = (sts,xs> + 2<gs.xs> + ms.

where m: E £2(O,m;H) is the solution of

m _ m -1 il 11
- 2<g D. R. :1D. H. D. R. 1P. xd > + 2<g. D 1Ri 1D* x >
i+1' i i i i di i+1' di
+ <gi D. R N1D H. D. R. 1D* g. >- <g. D 1Ri 1D g. >
+l' i i i 1. 1+1 i+1' i+1 '

. co _ eta-*1
Wlth Hi — Ki+l(I + D. 1Ri D. iKi+1)

Proof: If s is any fixed integer in [O,w) and
xs = h E H.. then, using the same arguments in the proofs of
Lemma 3.4.2 and Corollary 3.4.3, it can be shown that the

transformation h 4 ps is continuous from H into H so

that we have
or, since 5 E [O,m) is arbitrary, we have

co co co co .
(3.5.24) pi — Kixi + gi V 1 6 [0,”).

where (x:,p:) is a solution pair of (3.5.20) and (3.5.21).

65
Thus, the same arguments in the proofs of Theorem 3.4.4,

Theorem 3.4.5 and Theorem 3.4.6 follows to complete the

theorem. Q.E.D.
Next we consider the time invariant problem, i.e., we

assume that D1 = D, M1 = M, Ri = R, Si = S, Pi = P, and

E. = E.
1
Theorem 3.5.10: If the system (3.2.4) and the cost
functional (3.2.5) are time invariant, then the feedback
operator is also time invariant, i.e., K: = K00 for all

i E O, and Km is the solution of the algebraic Operator

equation

co

* a: -1
(3.5.25) K = ('3 K (I + DR

* _
DK”) 1®+ 1‘.

Proof: Let Xd = O . Since the system (3.5.20) and
(3.5.21) become autonomous, it is independent of initial
time s in the sense that if we translate the origin such
that i 4 i-s then p: = p: . Hence from (3.5.24) with

O)

m = 0 we conclude that K: = K . Q-E-D-

98 0

Thus we have completed the study of DTP on the
infinite time interval. The important results remaining is
the convergence properties as N 4 m.

we denote uN and uOD be the Optimal controls on the
intervals [O,N) and [O,m) respectively.

Theorem 3.5.11: Let ”Q” < 1 . Let 3N(§N,BN resp.)
be the extension of uN(xN,pN resp.) on [O,w) by 0 out-

side [O,N). Then as N 4 w,

‘l:

ll..l’\I I

66

(3.5.26) EN 4 u°° weakly in £2(O,°°;U) ,
(3.5.27) §N 4 xco weakly in 22(O,w:H),
(3.5.28) 5” .. p°° weakly in 22(o,oo;H) ,
(3.5.29) 8h 4 K:h weakly in H, V s E o, V h E H .

Proof: Let = inf JN (v), jco = inf Jm(v). For

jN
v E 12(O,m:U) we have, 1JN (v) 2S.Jm (v) and hence jN.S.jd

N N- AN
Thus jN = JN(u ) 2_C .Z) Hug ”2 and if u is defined as
1%

in the statement of the theorem, we have

(3.5.30) HEN“ 2 _<_ c .
1' (O,°°7U)

But then due to Remark 3.5.4 and (3.5.18),

(3.5.31) ”SEN” 2 gc,
2 (0.”:H)

(3.5.32) HpNH 2 gc,
I. (O,°°;H)

and again by virtue of (3.5.31),

(3.5.33) 11x3)! 3 c .
Hence we have from (3-4-3)

* J J N .
(3.5.34) Pi - ®ipi+l + I‘ixi + ei - (I‘NxN+eN)6(1-N) ,

(3.5.35) = @321:I - D. 1R; 1.1) :pi+l+?N- (@Nx11:+fN)6(i-N) ,

J
where 6(0) = l, and 6(r) = O for r # 0,, and ei and

 

The C's denoting constants independent of N.

67

fi are extensions of e. = — S.x + M.R—1P x and

1 1 d. 1 i i d.
1 1
fi = D. 1R'11Pixd on [O,N] respectively.
i

we may then find a sequence Nn 4 m such that

N
fi'n 4 u weakly in 22(0,m;U) ,
N N

x -v x , p 4 E weakly in 22(O,oo;H) .

Thus (3.5.34) and (3.5.35) become

_ *_._ _
(3.5.36) pi — @ip1+1 + F1 1 + el
- _ - ;.1

By comparing the above with (3.5.19), (3.5.20) and

co -- co

(3.5.21), we deduce that 5': and x = x, . The
P
relation
N _ -l * N -1 * N
(3.5.38) ui — R.l D. ipi+1 — Ri Mixi + RiPixd.

1

gives us in the limit (if necessary, take a subsequence),

(3.5.39) 51 = — R. _1D._ p. - R.fi1M lxi + R. 1P. 1x
1 i+l di

00

hence we have, by comparing with (3.5.19), that 5': u
Thus we have proved (3.5.26), (3.5.27) and (3.5.28).

To prove (3.5.29), we observe that th is defined by

N _ -1 N .
xi+1 — ®ixi - D. lRi D. ipi+l 1n [5;N),
(3 5 40) pN = 69.1pr xN in [s N)
' ' i i+1+ i i ' '

68
and then

_ N
th — p .

S

But (xN,pN) corresponds to the optimal control of

a system whose state is given by

xi+1 = @xi + Diui 1n [s,N), XS =‘h,

and whose cost is given by

N-l
JE'NM = i§S[<xi.c2ixi> + 2<X1'Miui> + <ui'Riui>]

By (3.4.22), we have

inf alsl'N(U) = <K§hrh> S Jlsi'N(O) S CHhHZ '

hence
(3.5.41) HKEhU S CHhM , C = constant independent of s and N.

Now if wN is optimal control of this problem, we obtain

from (3.5.30)
N-l 2
_2 lefu s c .
1=s

and extending wN by 0 for i 2 N, ‘we deduce that
~N “N
w (resp. 2“, p ) ranges in a bounded set of

22(0,m;U)(resp. £2(0,m;H)). we may then find a sequence
N N
Nn-ooo suchthat wn-ow,§n-ox,p’n-bp inthe

corresponding weak topologies and hence satisfies (3.5.19),

on — on

(3.5.20) and (3.5.21). Hence §=x , p=p and
N
P = P00 and psn 4*ps ‘weakly in H, completing the proof.
Q.E.D.

69
3.6 THE RICCATI INTEGRO-DIFFERENCE EQUATION

The Optimal feedback Operator K.1 is found to be the
solution Of the Riccati Operator difference equation. But
there are no straightforward procedures for solving
Operator equations directly. In this section we derive an
equation from the Riccati Operator equation which can be
solved analytically or numerically. This will be done by
showing that Ki can be represented by an integral Opera-
tor, and thus an integro—difference equation will then be
derived for the kernel of this integral Operator. Through-
out this section we choose H to be L2(D).

Theorem 3.6.1: The Optimal feedback Operator

Ki 6 £4L2(D):L2(D)) has a unique kernel Ki(z,§) such that

(3.6.1) Kix = f Ki(z,g)x(g)dg v x e L2(D).
D

To prove this we need the following theorem, so-called
Schwartz Kernel Theorem.
Theorem 3.6.2 (Schwartz Kernel Theorem, [S—l]): If H1

and H are locally convex spaces and T is a continuous

2
linear Operators from Hl into H2,. and if the following
are true:
. m I _ . _
(1) CO(D) C Hi c Hi 1 3(D), 1 — 1,2,
0 I m 9 o
(11) CO(D) 1s dense 1n Hl n H2,

then L can be represented by a unique integral Operator
whose kernel L(z,§) is a distribution on D x D.

Proof Of Theorem 3.6.1: For any i E o, Ki is
bounded linear Operator from L2(D) into itself, implying

/\

70
that K1 is continuous. By (2.1.1), c:(D) C L2(D) C fi(D),
and c:(D) is dense in L2(D). Thus by Theorem 3.6.2
there exists a unique kernel Ki(z,g) satisfying (3.6.1)
for all x E C:(D), and since C:(D) is dense in L2(D),
(3.6.1) holds for all x E L2(D) ‘with the limiting arguments.
Q.E.D.

Before we derive the Riccati integrO-difference equation

‘we define Operators Li’ Hi 6 £(H:H):

(3.6.2) L 1

.. *
o DCRI Dc '
1 l 1 1.

1

(3.6.3) H

i Ki+l(I + L

iKi+ 1)
Theorem 3.6.3: The kernel Ki(z,g) corresponding to the

Optimal feedback Operator Ki 6 £(L2(D), L2(D)) characterized

in Theorem 3.4.4, by (3.4.11), satisfies the integro-

difference equation

*

(3.6.4) Ki(z.g) = 61,261,, Hi(z.g) + Fi(Z.C).
(3.6.5) Ki+1(z.o = Hi(z.g) + ID(gagzmmiwmmiflw.Odo dp.
(3.6.6) KN(Z.€) = F(2.§) .

where Li(2.0). Hi(Z.P). Ti(2.C). F(2.C) and Ki(Z.C) are
the symmetric kernels corresponding to Operators Li' Hi' Ti,

* *
F and Ki respectively, and ®i is the Operation Of ®i

,2
on the argument 2 .

Proof: Since all the operators Li' Hi’ Ti, F and K

are bounded on L2(D), by Theorem 3.6.1, they have

71

corresponding integral representation whose kernels are
distributions on D x D., The Riccati equation for

x 6 L2(D) is from (3.4.11)

-1

_ -1
(3.6.7) Kix ®*iiiK+1(I+DRi D*i.K

1+1) @ix+I‘ix;KNx = Fx,

or equivalently, combining with (3.6.2) and (3.6.3), is a
system of equations
*
K.x = ®.H.®.x + F.x,
1 1 1 1 1

H. 1x + H. L. K.

(3.6.8) K. i i i+1x

X
ll

KNX = F}{.

Note that, since the kernel H. (2, Q) is a distribution,

H.®.x
1 1

(1311526611,, X(C)d€ = 63.52.3693."g x(c» sz)

= <6;g Hi(z.c).x<§)> 2 = f a: g Hi(Z.C)X(C)dC-

L(D) D '
Thus, applying the integral representations to (3.6.8), we

Obtain

(Dix (z. C)X(C)d€ = ID @"5 (z. C)X(C)dC+I ri(z.c)x(g)dg.

i, z ®i, i
QH D

(3.6.9)
iji+1(z.Ox(g)dc= IIIH (z.p)L (p o)1< 1(0 anodgdodp

D D D

+ [Himo X(C)dC.
D

ij(z.c)x(c>dc = I F<z.ox(c)dc.
D D

l P ‘I -(.Il'. ‘1! .

72
Since (3.6.9) holds for all x E.L2(D), we deduce (3.6.4),
(3.6.5) and (3.6.6). The symmetry of kernels comes from

the self-adjointness of Operators. Q.E.D.

Boundary conditions can be specified for the Riccati
integrO-difference equation (3.6.4) - (3.6.6). Let y be
the value of the transformation (3.6.1), then the evaluation

Of y at z 6 6D, the boundary, is

Y(Z) ZEaD =IK1(Z'C)|2€BD x(§)d§ °

D

 

Thus, the Dirichlet boundary condition, i.e.,

 

 

k

1% SD = 0 , 0 g k g m-l , gives us a condition for K. (2, Q):
an 1
KNZ'C) lzean = '53 Ki(z'€) 2631) = = anm- K1(2' ‘3) zEBD = 0'

where n is the outward normal at the boundary bD.. More-
over, by the symmetry Of Ki(z,C), the above boundary condi-

tion also holds for Q 6 8D, i.e.,

8 am-1
flm‘)fiw=33%“&)fiw=

 

1 Ki(z'§) gean = 0'

With the similar arguments in the derivation of the Riccati
integrO-difference equation, we can also derive an integro-

difference equation for gi characterized in Theorem 3.4.4,

by (3.4.12), i.e.,

gi(z) = ®:gzgi+1(z)- f ch3 zHi(z.g)Li<g.o)gi<o)do d;

i

+ ID£®i'zHi(z.C)Gi(g,o)xd (0)610 dg

(con't. on next page)

73

(con't. from previous page)

(3.6.10)

+ f wi(z.g)xdl (Odg.
D i

gN(z) = — J;F(z.c)de(c)dg.

where Gi(g,o) and Wi(z,g) are the corresponding kernels

Of bounded Operators G. = D.R71P., and W.==M.R71P.-S ,
1 l 1 1 l 1 1 1

1
respectively.

In sequel we may also represent the Optimal cost in

terms of the feedback kernel. Recall that the Optimal cost

is given by

(3.6.11) J = (K x ,x > + 2<g ,x > + m .
O O O L2(D) O O L2(D) O

where $1 is the solution of (3.4.23). Using the integral
Operator representation for K1' and evaluating the inner

products in L2(D), we Obtain

J = I f KO(Z.C)XO(C)XO(Z)dC dz
D D

(3.6.12)

-+ 2]. gb(z)xo(z)dz + Q0,
D

where mi satisfies the difference equation

0 . 'k
+ J J IIGi(Z.P)Hi(P.G)Gi(O,Q)xd (9xd (z)d(,d0dp dz
D D D D i i

(con't. on next page)

74

(con't. from previous page)
D D 1
Li

(2 Oxdi (Qg. i+1(2)dcdz

£1.52. C)Hi(P.O)Li(0. c)gi+1(c)gi+1(z)d<; do dp dz
1.(2 C)gi+1(C)gi+l(2)d€ dz .

cpN(z) = J'DfDF(z.c)de(g)de(z)dg dz .

with Yi(z,C) to be the corresponding kernel to the bounded
* -1
The results Obtained in this section can also be true
for the problem on the infinite time interval. Moreover, if

the prOblem (i.e., the state equation and the cost func-

tional) is time invariant, then we have the following set of

equations
(3.6.13) K(z.§) = 6:63: H<z.g) + 112.6).
(3.6.14) K(z,§) = H(z,p) +J' j H(z,p)L(p,c)K(c,g)dodp.

D D

CHAPTER IV

SAMPLED-DATA POINTWISE CONTROL

In this chapter the results Obtained in Chapter III
are specialized to the pointwise control problem, where
the control is applied only at a finite number Of points
in the spatial domain. The pointwise control prOblem
(PCP) is defined in Section 4.1. In Section 4.2 the
Optimal pointwise feedback control is derived, and it is
shown to be computationally simpler than the general feed—
back form of Chapter III. Section 4.3 deals with the
approximation of PCP), where a suitable choice of state
‘weighting Operator results in the finite dimensional
approximation by an eigenfunction expansion. An example

for PCP is considered in Section 4.4.

4.1 POINTWISE CONTROL PROBLEM (PCP)

In this section we shall formulate the pointwise
control problem, where controls are applied only at a fixed
number of points within the spatial domain of the system.

we suppose that control is concentrated on some finite
number Of spatial domains in D, say, El,:°-, EM,, and
the control is respectively constant with respect to z in
each domain. Here, we introduce the characteristic

function Xk(z) to each Ek defined as

75

76

1, 263k:
0, ng-kl

k _
(4.1.1) x (z) — k = 1' 2"”, M .
When the measure of Ek approaches to zero (in the sense

of Lebesgue), we Obtain the ideal case such that

(4.1.2) xk(z) = 6(z-zk), 2k 6 D, k = l,2,°°°. M-

where 5(2) is the Dirac's delta function in RP . NOte

that the control space U to be considered is M-dimensional
Euclidean space RM, and the control is M-vector ;3.

Definition 4.1.1: The pointwise control Operator D2,
defined for all i E 0 on RM, is as following:

M,
(4.1.3) D? u. = Z3 xk(z)d].‘ 11]? V E. 6 RM ,
1'—1 k=1 1 1 1

where xk(z) is given by (4.1.1), and for all i E 0, d? E R.

Strictly speaking the operator defined by (4.1.3) should
be called as a spacially concentrated control Operator, but
since later on we will approximate the results Obtained to
the pointwise case, we prefer to call it as pointwise control
Operator.

In order to apply the results Of Section 3.6 we require

that D? to be a bounded linear Operator from RM into

 

12(1)).
Lemma 4.1.2: For each i e o , D‘i’ is a bounded linear
Operator from RM into L2(D).
Proof:
D° 2d — 21“) k( )dk k)2dz
f(igi) Z‘J‘( Xziui

D I) k=1
(con't. on next page)

77

(con't. from previous page)

M 2 M 2
Z xk(z) (d).( u’?) dz = E (d).( 11‘?) xk(z)dz
k=1 ID 1 l k=1 1 1 ID

Since I x3(z)dz = u(Ek), the Lebesgue measure of Ek'
D

and since this is less than u(D), ‘we have

M 2
27 (b1; u?) = u(D) HQ

2
I (D? 11.1) dz 3 MD) isillzM .
D k=1 R

where 21 is the M x M diagonal matrix with entries
kk k '

Di = di , k = 1,2,°°°, M.. If ”QiHRM is the induced
matrix norm of 'Qi' it follows that D? is a bounded
linear Operator from RM into L2(D). Q.E.D.
Remark 4.1.3: If the pointwise control Operator
D: is defined in terms Of delta function (4.1.2) instead
of the characteristic function (4.1.1), then D? Bi 6/L2(D)
since the delta function is not square integrable, and
thus the theory of Section 3.6 is not applicable.
Since the control space is RM,. it remains to revise
the cost functional in this framework. Furthermore, for
the attraction Of simplified results we formulate a simpler

cost functional than the original one, (3.2.5), in DTP,

i.e., the cost functional for the pointwise control is

defined by
N-l
(4.1.4) J = i§0[<xi'Qixi> + <gi.§_it_1_i>] + <xN.F xN> .

where Ed is a symmetric, positive definite M x M. matrix
for all i E O, and, Qi and F are as defined in

Section 3.2.

78

NOW’We define the pointwise control problem (PCP):
Pointwise Control Problem (PCP): Given the discrete-
time system (3.2.4) with Di = 13‘; , defined in (4.1.3),
and the cost functional (4.1.4). Find a sequence of
* M

*
controls 3, = {Bi 6 R , i 6 a} such that for all

2,: £31 6 RM, i E o}

J(uf) = inf J(u) .
u

4.2 THE SOLUTION OF PCP
In this section the pointwise control problem (PCP)
is solved by applying the results of Section 3.6, that is
the feedback integral operator and the Riccati integro-
difference equation. If we apply the results of Chapter III
to the pointwise control prOblem (PCP), which has the
M

control space U = R and the pointwise control Operator

D: defined in (4.1.3), then the optimal control is given by

-1
1+1) 6 Xi '

* *
-l O

1 O 0
Di Ki+1(I + Di’i‘i Di K

* -
(4.2.1) Bi = :31

where Ki is the feedback integral Operator whose kernel

Ki(z,§) satisfies the Riccati integro-difference equation

*

Ki(z,§) = 9* 4g Hi(z.§) + Qi(Z.C) .

Z

(4.2.2) Ki+1(z,C) =Hi(Z.C) + jgjl'DHi(z.p)L‘i’(p.o)Ki+l(0.Ododp .

181(20 g) = F(Z,Q) a

*
where L:(p,0) is the kernel of the operator L2==Diggldi

79
we will now simplify the above equation and hence
the Optimal control (4.2.1). The adjoint pointwise control

*
Operator D? G £(L2(D): fin) is Obtained as following:

If y €L2(D) and Bi ERM, then

*

0
<1). y,u.> = <YID _ 11>
1 1 RM iu i L2(D)
M k
= I y(z) Z} xk (z)dk iui dz
D k=1
= :4? [I y(z) xk(z)dz]d].< u]? ,
k=1 1) 1 1

*
thus B? y is considered to be a vector in RM such that

. 1

(4.2.3) D‘i’ y ka(z)y(z)dz .
D

ll
0..
I-" 7?

where the bracket denotes a M-dimensional column vector.

The kernel L:(p,o) of the Operator L9 = Digngi* is

then Obtained as following: Using (4.2.3) we deduce

1 63-: I xk(0)y(0)do

D 1
t

M k k
=D. 2312.;l (j, k)di J‘ x (O)y(o)do
k=1 t D

o __ O -

l-‘O

M -1 k k
= 27x3 (p)d3 E R. (j.k)d. Ix (o)y(o)do
j=1k=1 1 l D

M
f [.231 k21x3 (p)d3a'.'1(j Md‘; xk (onwomo.
D 3= =

80
-l . . . th . -l
where Ri (j,k) 15 the 3k entry of matrix Bi . Thus

the kernel of Liy is given by

M
(4.2.4) L3(p.o> = 21 13:31 x 3(p)d3RT 1(j k)d1.‘x1‘(o).
j-

The double integration term in (4.2.2) may now be written

as

£IDH3(z.p)L3(p.o)Ki+l(o.Odo dp

(4.2.5)

= I I H (z, p) 23 23x3 (p)d3n'i' 3(j,k)d3 xk(o>xi+1(o.g>dodp
D D j=l k=1

M M .
= 23 Z[d 33ij (pm. (2 p)dp]R3 1(3' k)[d3 IDkx (WK-1 +'1(‘1 Odo] .
j=l k=1 D

and if we define vector functions hi(z) and ki+1(z) to

be
(4.2.6)
321(2) = d]; ,ka(p)H-1(z.p)dp .l<_i+l(z) = d3 ka(p)1<i+l(z.p)dp .

D 1 ' D 3
then (4.2.5) becomes hg(z)§;1 ki+1(§), where T denotes

the transpose. Thus the Riccati integrO-difference (4.2.2)

becomes
* *
Ki(z.o = 92 23 Hi(z.o + Qi(Z.C).
(4.2.7) Ki+1(z.o = Hi(z.c) + h3 (2)12; 1k3+1(c> .
KN(ZoC.) = F(Z, C.) o

and the Optimal pointwise control (4.2.1) becomes

= - Ri n° fDR. (z mgx (OdC

= - RilD°*

6* 1(2 C)X (€)dC

1 1£DH i
dkju *1
d1 IDX kw Hi (2. §)xi(§)dg dz
(4.2 8) 1
331 * k T
= - N§g(dJ x (z)H.(z,g)dz)xi(§)d§
I

D 31

=-RT1
“'1

Q
D

n!!-

So far, because of the difficulty stated in Remark 4.1.3,
i.e., DE 31 Z L2(D), 'we have been forced to use the charac—
teristic function (4.1.1) rather than the delta function
(4.1.2) in order to apply the theory Of Chapter III. Thus,
as a result, we have Obtained the equation (4.2.7) for which
we know a solution exists. Since we are interested in the
pointwise control (even though it is impossible to apply ideal
point source), we may replace the characteristic function
(4.1.1) with the delta function (4.1.2) in order to solve the
equation (4.2.7) approximately (cf.[P—4], [M-l]). Thus, if
we substitute (4.1.2) into (4.2.6) we obtain

(4.2.9) him) = d]; H. (z, z k) ,1_<i+1(z) = d1; Ki+1(z,zk) ,

1 t

and the optimal control (4.2.8) becomes

82

.353:
I
[.1

, *
= -R. Jug; hi(€)xi(€)d€

(4.2.10) = -R. j 2: d3 Hi(§,zk)xi(§)d§
D 1

i(€)xi(C)d€

m»

where Q. is M x M diagonal matrix with entries

kk R th

the k control coefficient in (4.1.3), and

£1 and Xi may be called the weighting_and output M-

vectors of the averaging device respectively such that

t

1

Note that it is necessary to determine the M functions

(4.2.11) .2: (C.zk) .

A A *
1 = I himximdg. him = gg H

D

Hi(z,zk), k = l,2,°'°, M, in order to completely specify
the Optimal feedback control. In the computational point
of view this is certainly simpler than the computation of
the entire kernel, i.e., the computation of Hi(z,§) for
all (z,§) E D x D..

we illustrate the feedback structure thus Obtained in
Figure 4.2.1 where "---->" and " ==$>" indicate the

flow of distributed quantity and M-vector respectively.

83

 

 

 

 

 

 

 

* ,
fi(z) =1): 1-1-i CONTROLLED SYSTEM xi(z)
r xi+1(z)=§xi(z)+fi(z) 1
g I
I I
I I
' I
' I
| I
I POINTWISE u* ,
I CONTROL -i Xi I . , .
L__dOPER2TOR 1 -gi 23 3 J J hi(z)[.]dz J
13.1 I D

 

 

 

 

 

Figure 4.2.1 Optimal feedback pointwise control system.

It is interesting to note that this feedback structure is
analogous to that Of continuous-time problem (cf.[G-1J).

we may conclude here that Obtaining a solution for the set
of functions {Hi(z,zk)}k:1 enables one to design appro-
priate instruments (not necessarily physical devices: could
be computers, managements, etc. ...) with weighting func-
tions equal to $2 Hi(z,zk).

Let us consider the pointwise control problem (PCP) on
the infinite time interval. If Hypothesis I, given in
Section 3.5, is satisfied, then an Optimal control exists for
PCP according to the results in Section 3.5. Moreover, if
the system is time-invariant, the time-invariant feedback and
weighting kernels are K(z,g) and H(z,g) respectively, and
corresponding k(z) and h(z) are Obtained from (4.2.9).

Thus (4.2.7) becomes the time-invariant algebraic Riccati

equation

84

*

mm» = 2* 43 H(Z.C) + Q(z.g).

z
(4.2.12)

H(Z.C) + hT(z)g‘1I_<_(g) .

K(2.C)
and the optimal control (4.2.10) becomes

11* = - 3‘12 113311 H<c.z1‘)xi(odg

l

A
= - R _gjycmimdc.
D

(4.2.13)

The Optimal cost for PCP is given by
J = (K xo ,xo>

(4.2.14) = f f K(z,g)xo(z)xo(g)dg dz.
D D

we shall consider a special class of solutions of

(4.2.12) in the next section.

4 . 3 APPROXIMATION

The pointwise control problem (PCP) on the infinite
time interval will be considered in this section. The
focus of the develOpment is finding a method for Obtaining
a solution for the Riccati equation (4.2.12). The kernel
Q(z,g) is approximated by the eigenfunctions of system
operator, and as a result an algebraic matrix equation is
Obtained.

we rewrite the Riccati equation (4.2.12) here:

85
'k

t; QC mm» + Q(Z.C)

K(2.C)
(4.3.1)

K(Z.C) = H(Z.€) + hT(2)5'1t<o

where Q(z) and k(z) are M-vectors with ith components
diH(z,zi) and diK(z,zi) respectively, and the points
zi, i = l,2,--°, M are control locations in D.

It is assumed that the system operator 4* has a
countable number Of eigenvalues 11, i = l,2,-°-, and that
the sequence of eigenfunctions {wi} forms a complete
orthonormal basis for the space L2(D). Note that this
assumption is well satisfied when 6* is a compact and
normal or self—adjoint Operator (cf.[C-2,p.359],[P-3,p.4ll],
[P-S]). we choose the kernel of the state weighting

operator to be

(4.3.2) Q(Z.C) 5- Emmyo .

where Q. is an n x n positive definite constant matrix
and y(z) is the n-vector whose ith component is the
eigenfunction wi(z). It may be shown that the state
weighting Operator Q with the kernel (4.3.2) is positive
semi-definite, i.e.,

<Qx.x> 2 = I I X(Z)y_T(z)Q_w_(§) x(ng dz
L (D) D D

[ IszmTIzMzIQIJ y(OxIOdg]
D D

£9120.

86

where x_ is the n-vector whose ith component is
(xywi> 2 . Note that the Operator Q is only semi—
definite(:;en though the matrix Q. is definite, because
there exist nonzero vectors x E L2(D) which are ortho-
gonal to the subspace generated by the first n eigen-
functions, resulting in <Qx,x>L2(D) = O . Note that if
n 4 w, the kernel Qm(z,g) of a positive definite
Operator is Obtained.

Now we derive an equivalent matrix Riccati equation
from the kernel equation (4.3.1).

we assume that the Optimal feedback and weighting

kernels are Of the forms

(4.3.3) K(z,g) f(zmmg).

(4.3.4) R(z.g) = RTIszIo .

where ‘g_ and g_ are unknown n x n symmetric positive
semi-definite matrices. Substituting (4.3.2), (4.3.3) and

(4.3.4) into (4.3.1) we Obtain

13(2):; Rug) 3:132»; 43 mo + firms; RIO.

(4.3.5)

313(2)::on £T(Z)EE(C) + RT(z) 3‘12“) .

where 2(2) and bfiz) are the M-vectors with 1th com-

ponents

aim) = digT(2)§ x(zi). bi(z) = 3111(2)}: u(zi) .

87

The vectors a(z) and b(z) may be written in the form

(4-3.6) 51(2) = W _mz). 2(2) = 2353(2) .

where 2_ is the diagonal M x M matrix with ith entry
i' ' . . .

D 1 = D1, the 1th control coeff1c1ent (cf.(4.1.3)),

and E1 is the M x n matrix with ijth element

W:L3 = w3(z1). Since y(z) is the vector of eigenfunctions

*
of @z, 'we have

*
(4.3.7) {>2 y(z) = _I_\_V_I7(Z) ,
where A. is the diagonal n x n matrix with ith entry
A11 = 11, the ith eigenvalue.

Substituting (4.3.6) and (4.3.7) into (4.3.5) we

obtain the equivalent matrix equation:

(4.3.8)

Ix
ll
b>
[E

Q

Eith-

lm

+

Im :-
+

12%

IU C)

5:

Solving the second equation of (4.3.8) in terms of g3,
and then substituting into the first one, we obtain the

matrix Riccati equation:

1 -l

T 21w 1+3-

(mlm £=1§I1t12§

We note that this Riccati equation is associated with
the following finite-dimensional control prOblem:
ginite-dimensional Control Problem (FCP): Given the

n-dimensional system

_ T . n
(4.3.10) 51+1'A-1ii3'1-v- Q31 . 50 ER ,

88

and the cost functional

(4.3.11) J= Z[x?g_x. + u?Ru.],
. —1 —1 —1 ——1
i=0
. ~k * _ * M
find a sequence of control u_ = {33,1 e 0, Bi 6 R ] such

that for all g_= [33,1 6 O, 2i 6 RM} 6 £2(0.”7RM).

J(1_1_*) = inf J(u) .
11

It is well known that (4.3.9) has a positive semi—
definite solution 5_ if the system (4.3.10) is completely
controllable, and furthermore if the system is also observ—
able, then 5_ is positive definite (cf.[K-l], [L-6]).

The necessary and sufficient condition for the system to
be completely controllable is found elsewhere (cf.[K-l],

[S-2]), that is, the n x nM matrix Q. defined by

(4.3.12) g [£32 'AWTD: ————— 311143112]
I

is of rank n if and if only the system (4.3.10) is
completely controllable.

Note that the system (4.3.10) is precisely the nth
order eigenfunction approximation of the distributed param-
eter system (3.2.4) with D = D9 by the method Of
Galerkin (cf.[P-6], p.15), for example, the n-vector co-
efficient of the forcing term Dogi by the eigenfunction
expansion is

M . . .
<Dou. ,w} = E x3 (z) d3u3 y(z) dz
1 L2(D) In j=1 1

(con't. on next page)

89

(con't. from previous page)
M . . . M . . .
= Z djug f x3(z)w(z)dz = E w_(zj)d3uJ
i=1 D 3=

0

ll
28
U

(C

i
which is equal to the forcing term in (4.3.10).

The weighting matrix g_ can also be computed from
(4.3.8) using the obtained solution 5_ of the Riccati
equation (4.3.9). Thus, utilizing (4.3.4) and (4.3.7),
the Optimal pointwise control (4.2.13) becomes

* = _R‘ln j 9’; wT(zk)§E(C)X-1(C)d€

= -349 !T(zk)_li A v_v_.(g)x (QdC
(4.3.13) = -R_ 12113 1f_<ox3(g)dc = {1221 3. A 22.1.

D

where xi is the n—vector coefficient of xi(z) in the
eigenfunction expansion. Employing (4.3.3), the Optimal
cost (4.2.14) with initial state xo 6 L2(D) becomes

J = I I xo(z)1<(z.oxo(g)d;dz=fxo(z)y_T(z>dz1<_J'maxomdc
D D D D

(4.3.14)

where 50 is the n-vector coefficient Of xo(z). Thus we

90

have shown that by choosing Q(z,g) to be of the form
(4.3.2), both the Optimal control and Optimal cost depend
only on the first n coefficients of the state variable

in the eigenfunction expansion.

4.4 AN EXAMPLE FOR PCP
we illustrate the approximation scheme by an example.
Consider the one-dimensional heat equation with point-

wise control Operator given by

 

2
3x t z a x(tyz)
(4.4.1) ___1_)_) = + B u(t) , o z 1,
x(t,0) = x(t,l) = O ,

where B0 is a pointwise control operator. The system

 

2
Operator A = 352 is self-adjoint and the eigenvalues are
32
“i = -i2w2, i = l,2,---, with orthonormal eigenvectors
‘w1(z) =\/92 sin 1 w z . The semigroup generated by A is
given by
m -12W2t i i
(4.4.2) §(t)y = Z: e y w (2) ,
i=1

where y1 is the ith coefficient of y in the eigen-

value expansion (cf.[P-5]). The equivalent discrete system
is

_ O
(4.4.3) Xi+1 — 4 xi + D Ei'

where the eigenvalues corresponding to Q = 4(6) are

91

.2 2
Xi = e 1 F 6 , i = l,2,°°°, and D0 is the pointwise

control Operator corresponding to B0 in (4.4.1).

we suppose that we are applying two pointwise con—

trols at z1 and 22, and choose the state weighting

kernel Q(z,§) to be

_ 1 2 w1<o
Q(Z.C) - [w (Z)w (2)] g .

w2(C)

where Q is a symmetric positive definite 2 x 2 matrix.

The matrix K. (cf.(4.3.6)) is

(/ 2 sin wzl \/F2 sin 21rz1

1—1 2 2
(/ 2 sin vz (/ 2 sin sz

The controllability matrix g_ is the 2 x 4 matrix

0 l I 2 O 1 I 2
d151n wz d251n wz delsin wz dezs1n wz
§_=(/2

. l . 2 . l . 2
d1s1n 27rz dzs1n27rz xzdls1n21rz 12d2s1n27rz

It is easy to show that the rank of g_ is 2, thus the
solution of Riccati equation (4.3.9) is guaranteed. If

there is only one control at 21, we have

__ dlsin vzl xldlsin vzl
§=/2
d sinzvrz1 sin27rz1

1 12d

1

which is of rank 2 since 11 ¢ 12 . Note that in the two

point control case 11 and 12 need not be distinct.

92
To compute analytically we choose that Q = g = g =

IH

the 2 x 2 identity matrix. If we denote the matrix ;5

to be
k11 k12
L = .
k12 k22
and choose the control points z1 and z2 to lie symme-
trically about the midpoint z =«%, i.e., z2 = 1 - 21,

then the Riccati equation (4.3.9) yields the following

2A k2 + (1 - 2A - e'2325) k - 1 - o
11 11 ‘
2B k2 + (1 - 23 - e‘8326) k - 1 - O
22 22 ‘
k12 = 0'

where A = 2 sin27rz1 and B = 2 sin227rz1 . Thus we obtain

 

 

k12 = 0'
-2w26 2 1 -2W26 -4w25 4 1 . 2 1
k _ -l+e +4sin wz + l+2e +e +l6sin W2 +851n vz
ll —
8 sin21rzl
(4.4.4)

 

2 2 2
k _ -1+e-8W 6+4sin221r231'3I-y/1-f-2e—87r 6+e-16W 6+l6sin421rzl+881n227rzl

 

22 —
8 sin221rz1
The optimal cost (4.3.14) becomes

(4.4.5)

J(xo) = I £)xo(z)[kllsin vz sin w§+-kzzsin 2wz sin2wg]xo(g)dgdz
D

93

The weighting matrix g_ is Obtained from (4.3.8), i.e.,

 

 

 

 

(4.4.6) 5 = [1&3 - (gt/f1 ,
2w26
which yields the entries h11 = e (kll-l),
2
_ 8W 5 _
h22 — e (k22 l), and h12 — O . Thus, from (4.3.13),
the Optimal control is
sin W x.( )d
3 91 92 ID 4 l c c-
(4.4.7) 111 = ,
g1 —92 I sin 2ng xi(§)dg
D
where
-2w26 2 1 -2w26 -4n25 4 1 2 1
-l+e -4sin WZ + 1+2e +e +l6sin vz +83in wz
91 - -w26
4e sin wz
2 2 2
- 1 _ -
-l+e 8” 6-4sin221rz +v/1:2e 8” 6+e 16” 6+l6sin421rzl+8sin227rzl
9 = -
2 2
4e-47 6sin szl

Note that if we have the measuring devices which yield the

output
J‘sin WC }L(§)d§
D l
11 = '
I sin ZWQ xi(g)dg
D

then yi can be fed directly through the gain matrix

g1 92

IO
fl

91 ‘92

to Obtain the Optimal control

It should be noted that the measurement does not depend
on the control point locations, and only the gain matrix
§_ does. Thus the measurement and control problems are
decoupled and the changing control point location does not
alter the structure of measuring devices. This provides
us the choice Of control point location which minimizes the
average cost in some sense. The system in the example is

illustrated in Figure 4.4.1.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

f(t,zz) _ CONTROLLED SYSTEM x(t,z) xi(z)
2 ....... - .. ..

f(t,z1) g—f- = 11—323 + f(t,z) b SAMPLER '3

dz 3

I

I

I

* 1 I

.13 31 '

J O-ORDER 1 If sinw§[-]d§ 3

HOLD I D I

Q 2 ' I

 

 

 

 

 

 

 

- 2* y, i .
31:23:? l—Iul - 1 Ir “WIMP
- D

Figure 4.4.1 Sampled-data control system for Example 4.4.

 

 

The Optimal control point location can be found analyt-
ically as a parameter Optimization problem. The Optimal

control is given by (4.3.14), i.e.,

J(X)=XTKX o
"O “0“‘0

where x0 is the n-vector coefficient of initial state xo(z).

95

If we assume that is uniformly distributed over the

x
_0
unit sphere |x.| = 1, then it can be shown that

O
(cf.[K-3]) the average cost of J(xa) over the unit

sphere is

= 3'- trace (5) = 3.2-(R +k

Javg 2 11 221'

where kll and k22 is given by (4.4.4). Thus the Op-
timal point location of z1 (and 22) can be found simply
by differentiating Javg and setting it equal to zero.

We have completed the example in this section. Now we
summarize the results of approximations for (PCP, i.e., a
suitable choice of the state weighting kernel Q(z,§)
yields the finite dimensional approximation by eigenfunction
expansion, from which we conclude that the optimal control
law which feeds back only the first n modes under consid-
eration is Optimal over the class of all feedback control
laws. In the example it was shown that the feedback struc-
ture of the PCP can be separated into a measurement part,

which is independent of control point location, and a gain

part, which depends on control point location.

CHAPTER V

SUMMARY AND CONCLUSIONS

This dissertation is concerned with the sampled-data
control Of distributed parameter systems with quadratic cost
criteria, where the system Operator is an infinitesimal gen-
erator of a strongly continuous semigroup Of bounded linear
Operators. It is shown that the Optimal control exists and
is given by a bounded linear transformation Of the sampled
states of the system. The resulting Optimal feedback opera-
tor is shown tO be the solution of an operator difference
equation of the Riccati type. The feedback Operator is
represented by an integral operator whose kernel satisfies
an integrO-difference equation.

These results for general infinite dimensional control
problem are specialized to the problem of pointwise control,
where control is applied only at a finite number of points in
the spatial domain. The pointwise feedback control is shown
to be Of simpler form than the distributed feedback control.
It is also shown that a particular Choice of the state weight-
ing Operator yields a finite dimensional approximation by the
method of eigenfunction expansion, and a decoupling of the

measurement and control prOblems.

In summary, this dissertation creates a new problem of

infinite dimensional discrete-time Optimal control, gives a

96

97
simple approach to solving control problems of this type
and generates new results for infinite dimensional systems.
This approach enables us to make direct extensions of the
results for finite dimensional systems to infinite dimen-
sional discrete—time systems.

There are a number of topics for further research

based on this work, for example:

(1) The feedback control law obtained requires the
distributed state x(z) over the entire spatial domain.
However, in many cases, the state is observed only at a
finite number of points. Thus the pointwise output feed-
back problem analogous to that of finite dimensional systems
(cf.[L-7]) should be investigated.

(2) In Section 4.4 the Optimal control point location
was briefly studied. This problem can also be generalized
to a problem with an arbitrary set of control points by
minimizing the averaged cost in some sense over the set of
possible control points.

(3) The computation of solutions to Riccati Operator
equations might be developed applying the Galerkin technique,
successive approximations (cf.[P-6]), or steepest descent
methods (cf.[B-S]).

(4) The stochastic Optimal control of discrete—time
distributed parameter systems should be investigated. This
prOblem is analogous to that of continuous-time distributed

parameter systems studied by Bensoussan [B-7].

98

(5) The variational approach might be used in the
identification of discrete-time distributed systems in a
manner analogous to that of continuous-time systems
(cf.[P-6], [B-6]).

The prOblems (4) and (5) are under investigation by

the author and will be reported in a subsequent work.

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APPENDIX

APPENDIX

PROOFS OF LEMMAS 3.3.6 AND 3.3.7

Lemma 3.3.6 (Lions, [L-l]): If W(u,v) is a continu-
ous symmetric coercive bilinear form, then there exists a

*
unique u E‘V such that

C(u*) = inf C(u) .

u€V
Proof: (Existence). Let un G'V be a minimizing
sequence, i.e.,
(A.l) C(u ) 4 inf C(u) .
n
u€V

Note from (3.3.6) that C(u) 2.CHuH2 - ClHuH, thus with
(A.l) un is bounded. Therefore we can choose a sub-
sequence uk of un which converges to a weak limit, say,
w €‘V.. Since w(u,u) is lower semicontinuous and L(u)
is continuous in the weak topology of V, C(u) in (3.3.6)

is lower semicontinuous and inf C(uk) 2_C(w). Applying

(A.1) we have inf C(u) 2_C(w), w €,V . Hence it is neces-
u€V *
sary that C(w) = inf C(u) and u = w.
uGV
(Uniqueness). The function v(u,u) is strictly convex,

because for 1 6 (0,1), ul, u2 €‘V,

104

105

W((1-1)u1 + 1 u2, (l-X)u1 + 1 u2)

= w(ul+1(u2-ul), ul + k(uZ-ul))

2

= w(u1,ul) + 21F(ul,u -ul) + 1 w(u2-u1.u2-u

2 1)

< W(u1,ul) + 21v(u1,u2-ul) + 1 w(u2-ul,u2—ul)

=(l-X) W(u1,ul) + 1 w(u2,u2) .

Hence (3.3.6) implies C(u) is also strictly convex.
Suppose both ul and u2 assume the infimum of C(u)
on V', then V is convex (since it is a whole space)
. . l

1mp11es 2(ul-I-u2) 6 V and

C(%-(ul+u2)) < 3-0(u1) + 3-0(u2) = 125 C(u),
u

which contradicts the assumptions on 111 and u2 unless

Q.E.D.

Lemma 3.3.7 (Lions, [L-1]): If the hypotheses of

 

Lemma 3.3.6 are satisfied, then the minimizing element

*
u €.V is characterized by

(A.2) W(u*,V) = L(v) v v e‘v.

*
Proof: Suppose u is the minimizing element in V,

then
C(u*) g C((1—1)u* + 1w) v w e v and 1 6 [0,1]
or

'k 'k *
C(u + 1(w-u )) — C(u ) 2_0 .

Using (3.3.6),

106

* 'k * ~k * *
W(u + 1(w-u ),u + k(w-u )) - 2L(u +1(w-u ))

- [v(u*,u*) - 2L(u*)] 2_0
or

* * ~k * 1k
2[W(u ,w-u ) - L(w—u )] + 1 W(w-u ,w-u ) 2_0
As 1 4 0 we have
* 'k 'k

*
Since w is any vector in V', ChOOSlng w = u i_v, v E‘V

implies (A.2).
The necessity is proved by using the convexity of C(u),

i.e., for k 6 [0,1],
* 1 a: *
C(v)-C(u ) 2.1-[C((1-1)u + 1v) - C(u )] V v 6 V
Using (3.3.6) we Obtain
* * 'k * * *
C(v)-C(u ) 2_2[U(u ,v-u )-L(v-u )] + XTr(v-u ,v-u )
As 140 we have
* * * *
C(v) - C(u ) 2_2[w(u ,v-u ) - L(v-u )],

*
and thus (A.2) implies C(u ) g_C(v) V v 6 V .