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A CONTROL PROBLEM FOR BURGERS’ EQUATION

by

Sungkwon Kang

Dissertation submitted to the Faculty of the

Virginia Polytechnic Institute and State University in partial fulfillment of the requirements for the degree of

DOCTOR OF PHILOSOPHY

in

Mathematics

APPROVED:

Ook A. Lemar J. A. Burns, Chairman

tHe — K. B. Hannsgen

—L& Nur Qrrcer Rika Lo Wharde T. L. Herdman R. L. Wheeler

April, 1990

Blacksburg, Virginia

A CONTROL PROBLEM FOR BURGERS’ EQUATION

by

Sungkwon Kang

Committee Chairman: John A. Burns

Mathematics

(ABSTRACT)

Burgers’ equation is a one-dimensional simple model for convection-diffusion

phenomena such as shock waves, supersonic flow about airfoils, traffic flows, acoustic

transmission, etc. For high Reynolds number, the open-loop system (no control)

produces steep gradients due to the nonlinear nature of the convection.

The steep gradients are stabilized by feedback control laws. In this phase, suf-

ficient conditions for the control input functions and the location of sensors are

obtained. Also, explicit exponential decay rates for open-loop and closed-loop sys-

tems are obtained.

Numerical experiments are given to illustrate some of typical results on conver-

gence and stability.

ACKNOWLEDGEMENTS

First, I would like to express my deepest appreciation to my advisor, Professor

John A. Burns for his support, encouragement and understanding, to whom I am

in debt for constant motivations into new areas and ideas.

I also wish to thank Professors Joseph A. Ball, Kenneth B. Hannsgen, Terry

L. Herdman and Robert L. Wheeler for serving on my committee and their valu-

able suggestions. Thanks are also due to Professors Werner E. Kohler, Max D.

Gunzburger, Michael Renardy and David L. Russell, of Mathematics Department,

Eugene M. Cliff, of Aerospace and Ocean Engineering Department, William T. Bau-

mann, of Electrical Engineering Department, for valuable discussions and numerous

suggestions.

I would also like to thank members of the ICAM (Interdisciplinary Center for

Applied Mathematics) community for their friendship and kindness.

I wish to acknowledge the financial support I received from the National

Science Foundation under Grant ECS-8109245, the Air Force Office of Scientific

Research under Grant AFOSR-85-0287, AFOSR-86-0085 and AFOSR-89-001, the

Strategic Defense Initiative Office/AFOSR under Grant F49620-87-C-0088 and the

Defense Advanced Research Projects Agency under Grant F49620-87-C-0116.

Finally, special thanks are due to my lovely wife, Jeongook, my son, Michael,

and my mother in law, Hyou-Soon Yang, for their endless love and patience.

ill

TABLE OF CONTENTS

Abstract Acknowledgements

Chapter I. Introduction

1.1. Introduction

1.2. Physical Models

1.3. Previous Work Related to the Problem

1.4. Notation

Chapter IJ. Well-Posedness and Stability of Burgers’ Equation

2.1. Basic Definitions and Preliminaries 2.2. Well-Posedness and Stability of Burgers’ Equation

Chapter III. Linear Control Problem and the Exponential Decay Rate

of the Controlled Burgers’ Equation

3.1. Distributed Parameter Control Problem

3.2. Applications to Burgers’ Equation

Chapter IV. Approximation and Numerical Results

4.1. Approximation Schemes for Linear Regulator Problem

4.2. Feedback Schemes

4.3. Numerical Results

4.4. Summary

References

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Chapter I. Introduction

1.1. Introduction.

In this chapter, we present several examples to motivate the control problem for

Burgers’ equation. In Section 1.2, we consider some physical examples which can

be represented by Burgers’ equation. A brief history of the control problem related

to Burgers’ equation will be discussed in Section 1.3. In Section 1.4, we present the

basic notation.

1.2. Physical Models.

Many physical problems can be modelled, approximately or exactly, by Burgers’

equation. In this section we consider some of the more illuminating examples that

are available in standard references. Other examples involving acoustic transmission

and turbulence in hydrodynamic flows can be found in [14] and the references given

there. The following three examples are taken almost directly from [14].

Example 1.2.1. (Shock Waves).

An impulsively-started piston moving at a constant velocity into a tube contain-

ing a compressible fluid initially at rest creates compression waves. The compression

waves eventually coalesce, due to the nonlinear nature of the convection, to form a

single shock wave. The one-dimensional unsteady motion of the fluid is governed

by the continuity equation

a (t,x) + p(t 2) —-o(t xz) + v(t 2) 2 (t,z) =0 (1.2.1) ar? , p ? Or ? ’ az” ? _ “o-

and the z-momentum equation

< ult, xr) + v(t, ae v(t, z) (2 5 Plt, t))/p(t,z) = = ott x), (1.2.2)

where p is the density, v is the velocity, p is the pressure and 6 is the “diffusivity of

sound”. It is convenient to replace the density by the sound speed, a = a(t, z), via

a(t, z)/ao = (p(t, z)/ po) =, where ¥ > 1 is the specific heats ratio and the subscript

0 refers to the undisturbed values, ([27]). Equations (1.2.1) and (1.2.2) become

F alty2) + o(h2)Zalt,2) +1 *a(t,2)2o(t,2)=0 (1.28)

and

2

< ot r) + v(t, t) 7 Sot r) +4 a(t, tx ° a(t r)= 52 5q2 ht), (1.2.4) —1

where 6 is a function of the undisturbed (to the right of the shock) viscosity, density,

specific heat and thermal conductivity of the medium. Equations (1.2.3) and (1.2.4)

can be simplified by introducing the Riemann invariants,

_ a(t,z) v(t, 2) _ a(t,z) v(t, z) r(t,r) = y-1 + 9? s(t,z) = yo 1 5 (1.2.5)

to give

0 9? 35 qi (r(ts x) — s(t, z)) (1.2.6)

Nol SH

r(t,z) + (a(t, x) + v(t, ont r)=

and

0 0 & 3 3p o(te) — (alt, 2) — v(t, 2))a- s(t, 2) = sao (s(t 2) — r(t,2)). (1.2.7)

Consider the propagation of a disturbance into an initially undisturbed region,

$ = 89 where so = **.. Then the problem is governed by equation (1.2.6). But from

equation (1.2.5),

a(t,z) + v(t, 2) = —— ir(t, z) +2 5 3 5, (1.2.8)

thus equation (1.2.6) becomes

—-3 ,0 6 3 S(t xr) + (vt* L(t, x) 44 5 Soa er(t, zr) = 5 q2" tt x). (1.2.9)

As the final step we introduce the change of variables

y+1 u(t,z) = 5 (r(t,z) — ro), €=zr— aot, (1.2.10)

to give Burgers’ equation

0 6 By ult 8) + ule, as gull, = 358 u(t, €). (1.2.11)

From equations (1.2.8) and (1.2.10) we have u(t, é) = { a(t,€)+v(t, €) }—{ vo +a },

where vp = 0, ie., u(t,&) is the excess wavelet velocity (the difference between

propagation speeds of disturbance in stagnation and nonstagnation conditions). The

coordinate € is measured relative to a frame of reference moving with the undisturbed

speed of sound do.

Example 1.2.2. (Traffic Flow).

One-dimensional continuum model of highway traffic flow problem is governed

by the continuity equation

0 0 By ults zr) + 5g Mts zr) = 0, (1.2.12)

where u(t, z) is the density of cars and f(t,z) is the flux of cars, ([41]). If drivers

look ahead and modify their speed accordingly, then f(t,r) can be represented by

0 f(t, xv) = u(t, z)v(t, z) — Da ult, x), (1.2.13)

where v(t, x) = vo(1 — u(t,z)/u,) is the local mean velocity, vg is the speed when

u(t,z) = 0, u, is the saturation density at v(t,z) = 0, and D > 0 is the diffusion

coefficient given by

_ a? D=rv,, (1.2.14)

where v, is a random velocity of cars and 7 is the mean collision time for cars, ([30]).

Combining equations (1.2.12), (1.2.13) and (1.2.14) gives

(Sule t) + v9 3 u(t, 2) — 2( Oy Salt r) - Dut z)=0. (1.2.15) z=

By introducing the moving reference frame

£ = —2 + vot (1.2.16)

we obtain

8 vt, e) 42(2%)u(t,€)2-u(t,€) — Do-u(t,€) <0 1.2.17 3 ult §) + (ult Qagul »€) — pga lts §) = (1.2.17)

Let w(t, €) = u(t, €)/(S+). Then the equation (1.2.17) becomes

0 0 3 3p wt §) + vow(t, C)aewlt, g)- Daavlté) =0 (1.2.18)

Finally, we introduce the normalizations

t g sar and n= z (1.2.19)

to obtain Burgers’ equation

0 0 1 3 Be wis) + w(s, 1) qs) = Rog n); (1.2.20)

where 2p = Volo, to is the mean time between successive cars passing a stationary

observer and R is the dimensionless constant defined by

R= (22%, (1.2.21) Vr T

Note that the normalized reference frame 7 is given by 7» = —% + ig and that

u(t, £) and w(s,7) = u(¢ ~)/(%) represent the relative car density and the > r0

normalized car density, respectively.

Example 1.2.3. (Supersonic Flow about Airfoils).

For slender wings or airfoils subjected to a steady-state supersonic inviscid flow,

it is convenient to think of the wing as a perturbation of a uniform free stream. If

the governing partial differential equation is expressed as a velocity potential ¢(z, y)

it is possible to develop a hierarchy of solutions, the lowest order of which is linear.

In fact, the governing equation for the first-order problem is given by

2 2

(1 — M2) 42,0) + 5a0e0) = 0, (1.2.22)

where M,, is the freestream Mach number, z is the streamwise coordinate and y is

the stream-normal coordinate, ((14, p161)).

Although solutions to equation (1.2.22) give an accurate description of the flow

at an intermediate distance from the body, in the farfield a nonlinear second-order

correction must be added to account for the merging of the characteristics. In this

case, equation (1.2.22) is replaced by

(1 M3 - (7+ ML 262, Soe.) + Goley) =0, (1.2.23) oo Or ? Or? ? Oy? ? ? cave

where y > 1 is the adiabatic gas constant, e.g., for air, y + 1.4.

However, the description (1.2.23) is inappropriate locally when characteristics

merge the velocity potential, since a shock wave will occur. One way to permit

shocks to appear in the solution is to add a diffusion term to equation (1.2.23),

({9]). In this case, equation (1.2.23) is replaced by

5 62, y)+(1—M2,—(7+1) M4, 2-4(2,y)) 2562, y) +5 4(@,y) = 0, (1.2.24) 6x3 oY Co 7 ° Or Y Or? Y Oy? VY — 3 cate

where 6 > 0 is asmall diffusion coefficient. The term 6 42, y) will only contribute

to the solution in regions where shocks occur. In the limit 6 — 0 the inviscid solution

is recovered. The effect of including the term 6 ¢(z, y) permits a continuous

solution to be obtained in the vicinity of the shock so that the gross effect of the

shock on, say, the pressure distribution on the airfoil can be obtained.

For simplicity, consider a symmetric airfoil and introduce the following transfor-

mation of coordinates. Two directions appear in the problem, that of the freestream

and that of the outgoing Mach lines. The airfoil lies nearly along the former and

the wave pattern nearly along the latter. Thus the solution can be expressed more

naturally in terms of the oblique coordinates

—/M2—-—ly and n= JM2,-1y. (1.2.25)

Then equation (1.2.24) becomes

2 sa d6,n) - (y+ 1) MES 6n a 6(6,0) — 2(Ma, — 1) ame 7067)

ga? 2 —- = + (Mz, - 1) 973 $(€,n) = 0. (1.2.26)

With only single wave family oriented along the planes £ = constant, the term

(M2, -— 1) Z74(€ 7) is of higher order and may be dropped, ((20]). Hence, in what

follows we consider

sas (6s7)— (y+1) MS solr Sabb n)—2(M2,-1 \anae4 $(€,n) = 0. (1.2.27)

Finally, by the additional transformations

_ - ee an —-./xM —

Here, u(¢,) = ey #O(/f2(MZ —1)€,./M2, —1y) is the modified velocity

component in the outgoing Mach line direction.

1.3. Previous Work Related to the Problem.

The Burgers’ equation

0 0 0? 5 ult) + u(t, x) > pg ult, r) = ‘aa sult, r) (1.3.1)

was introduced by Burgers, ((5,6,7]), as a simple model for turbulence, where € > 0

is a viscosity coefficient. Since then many researchers have considered the conserva-

tion law

0 0 ay ult zr) + u(t, t) a u(t, z)=0 (1.3.2)

and the “viscosity solution”

u(t, 2) = lim u‘(t, x), (1.3.3)

where u‘(t,z) satisfies equation (1.3.1), ([10,19,22,26,28,29,31]).

Oleinik, ((31]), proved that for any L®-initial data, there is a unique viscosity

solution for equation (1.3.2) and the solution satisfies the “entropy condition”

u(t,¢ +a) ~ult,z) _ = (1.3.4) a

for all t > 0, a > 0, —oo < zx < oo and for some constant E > 0. A complete

discussion of these results may be found in [37].

Almost no results exist for the control problem associated with Burgers’ equa-

tion. Chen, Wang and Weerakoon, ([8,40]), considered an optimal control problem

for equation (1.3.2) with —co < x < oo. The problem was to select an initial func-

tion to minimize a specific cost functional J. They obtained sufficient conditions for

the differentiability of J with respect to the initial function and explicit expression

of the entropy solution of (1.3.2) in terms of initial data.

In this paper we consider a control problem for Burgers’ equation (1.3.1) defined

on a finite interval. Specifically, we will find several feedback laws stabilizing the

nonlinear system (1.3.1) with a certain exponential decay rate. The feedback laws

will be obtained from the linearized equation. From this point of view, Curtain,

({12]), considered a stabilization problem for semilinear evolution equations. Using

Kiehofer’s stability results for semilinear evolution equations, ([24]), she showed

that, under certain conditions, there exists a finite dimensional compensator which

produces a stable closed-loop system. These finite dimensional compensators are ob-

tained from the linearized control system. Applying her results to Burgers’ equation

(1.3.1) with, for example, Dirichlet boundary conditions, one can obtain stabilizabil-

ity results of the closed-loop system which are very similar with ours. However, in

this case, there is a restriction on the action of output operators. The domain of the

10

output operator should be taken as a certain proper subspace of L* which contains

the Sobolev space H2.

Throughout this paper, feedback laws will be chosen as “optimal”, instead of

finite dimensional compensators, in the sense that they minimize certain energy

functionals.

Well-posedness and stability results for the open-loop system are obtained in

Chapter II. In Chapter III, a “shifted” linear control problem ((LCP)*) is intro-

duced. Under appropriate selection of input and output operators, the (LCP)* is

stabilizable and detectable. The feedback control law obtained from (LCP)”* pro-

duces the desired degree of stability for the closed-loop nonlinear system (Theorem

3.2.10). Finally, in Chapter IV, a numerical scheme for computing the “feedback

functional gains” is developed and several numerical experiments are performed.

1.4. Notation.

If (X, ||-||x) and (Y,||-||y) are normed linear spaces, then £(X,Y) will denote

the space of all bounded linear operators from X to Y. For any A € L(X,Y), ||Al|

or ||A||ccx,y) will denote the operator norm on the space £(X,Y). In the event that

X =Y we denote £(X,Y) by C(X). From time to time we will use || - || without

any subindex for vector or operator norm. In all such cases the appropriate index

for || - || will be understood from the context. For a Hilbert space X, we denote

11

the inner product on X x X by <-,: >x. Given a linear operator A from X into

itself, we denote its domain, spectrum, resolvent and adjoint by D(A), o(A), p(A)

and A*, respectively. For real numbers a, b with a < b, L?(a,b;X),1 < p < oo,

will be the space of all Lebesgue measurable functions f from (a,b) to X such that

lf llze(a,b) = (f° [f(x)|Pdx)? < oo. The spaces H*(a, 6) and H#(a, b) are the standard

Sobolev spaces defined by H*(a,6) = {f € L?(a,b)|f) € L?(a,6),7 = 0,1,...,k}

and Hk(a,6) = {f € H*(a,b)| f(a) = f(b) = 0,7 =0,1,...,& — 1}, respectively.

The dual space H~*(a, 6) of H§(a, 6) is the space of all continuous linear functionals

on H¥(a,6) represented by the inner product < .,- >12(a,b):

Chapter II. Well-Posedness and Stability of Burgers’ Equation

In this chapter, we consider well-posedness and stability properties of the solu-

tion for Burgers’ equation with Dirichlet boundary condition. These results will be

needed in the analyses of our control problems in Chapter III.

2.1. Basic Definitions and Preliminaries.

Consider the initial value problem

£ a(t) + Az(t) = f(t,2(t)), — £ > to (2.1.1)

z(to) = 2

on a Hilbert space X, where —A is the infinitesimal generator of an analytic semi-

group S(t) satisfying ||S(t)||c(x) < Me’, t > to, for some constants M = M(w) > 1

and w > 0. Since S(t) is analytic, the fractional powers of A; = A+ al are well-

defined for any a > w, ([32, Chapters 1 and 2]). Since 0 € p(A,), the resolvent of

A, A} is invertible for all 0 < » < 1. Therefore, the graph norm ||z|| + ||A{z|| on

the domain D(A) of Af is equivalent to the norm ||z||,, = ||Ayz||. We denote the

Hilbert space D(A}) with the norm ||z]| + ||Afz|| or |[Ayz]] by (X,, || - lls).

We shall make the following assumption, (see [21,32]).

HYPOTHESIS (F): Let U be an open subset of [to,00) x X,. The function f :

U — X satisfies the hypothesis (F) if for every (t,z) € U there is a neighborhood

12

13

V CU and constants L > 0, 0 < 8< 1 such that

II f(t1, 21) — F (ta, za) Lx S L( [ti — tal? + lle — zell,) (2.1.2)

for all (¢;,2;) € V,¢ = 1,2, ie., f is locally Holder continuous in t, locally Lips-

chitzian in z, on U.

Now we are ready to state the local existence theorem for the solution of equation

(2.1.1). The following result appears as Theorem 3.3.3 in [21].

THEOREM 2.1.1. Let A be as before and f satisfy hypothesis (F). Then for any

(to,20) EU C Rt x X,, there exists T = T(to, 2) > 0 such that equation (2.1.1)

has a unique (strong) solution z(t) on [to,t9 + T) with initial value z(to) = 2p.

2.2. Well-Posedness and Stability of Burgers’ Equation.

In this section we consider the well-posedness and stability properties of Burgers’

equation, with Dirichlet boundary condition, on a finite interval [0,2] given by

ot r)= eH at zr) — z(t 2) 2-2(1 zr) 0<r<t, t>0 at ? Or? } } Ox 3 } }

z(t,0) = 2(t,6) =0, (2.2.1)

2(0,r) = z(z),

where € = «+ > 0, Re is the Reynolds number. In order to place the system (2.2.1)

into a semigroup framework let z(t)(-) = z(t,-), zo(-) = 2(0,-) and H = L(0, 2).

14

Define an operator A by

Ag = €¢" (2.2.2)

for all @ € D(A) = H?(0,2)N Ai(0, 2). Then the system (2.2.1) can be written as

an initial value problem

£ a(t) = Az(t)+ f(t,z(t)), t>0 t (2.2.3)

z(0) = 2

on the space H, where f(t,z) = —zz’ is defined on the space H}(0, 2). It is well-

known, (see [21,32,39]), that A generates an analytic semigroup S(t) on H.

We summarize the basic properties of the infinitesimal generator A and its

semigroup S(¢), t > 0, in the following remark.

REMARK 2.2.1. (i) The spectrum o(A) of A consists of all eigenvalues 4, =

—en*x?/£?,n =1,2,..., and for each eigenvalue \, the corresponding eigenfunction

gn 18 given by

én(t)=V2sinF2, O0<a<t (2.2.4)

(ii) The operator A is self-adjoint, i.e, A= A*, and the semigroup S(t) can be

represented by the following formula

S(t)z = So ell < 2, bn > bn (2.2.5) n=1

for all z € H, where ¢,’s are defined by equation (2.2.4). Moreover, from equation

15

(2.2.5), it is easy to see that S(t) has the stability property

IS@llean < ew, tO. (2.2.6)

A simple application of Schwartz inequality gives the following first Poincaré

inequality, ((42, p116}).

LEMMA 2.2.2. For any z € H2(0, 2),

llzll < 4llz'Ila, (2.2.7)

where H = L?(0, 2).

PROOF: For any z € Hj(0,2), z is absolutely continuous and is given by z(r) =

Jo 2'(s)ds, 0< @ < 4, and hence, by Schwartz inequality,

ees [ “|2(s)| ds

<( [ “ds )} / "|2!(s)/? ds)! L

< vi [ \e(s)Pas = VE

Therefore,

t 1

ella = ( [ l2(s) |? ds)?

< Vi llz'lla( / ds) Rl

= e|l2'lIx.

16

REMARK 2.2.3. (i) The above lemma gives an equivalent norm ||z||1 = ||z’||z2 on

the space Hj(0, 2).

(ii) It is well-known that D((—A)2) = H2(0, 2), ((21, p29],[25, p326)).

LEMMA 2.2.4. For any z € H}(0,£) = D((—A)?), the following inequalities hold.

(1) ]S@)ezllaa Se [Izllaz, = (#2 0), (2.2.8)

.. 1+21 1+), _ (i) USWellag $(FEGt Fe lellowm, > 9),

(2.2.9)

where y = ex? /€?.

Proor: For any z € H2(0,2), we know that (—A)?S(t)z = S(t)(—A)?z. Hence,

by Remark 2.2.1,

[IS(¢)zllag = IS (zllar + II(—A)? (zl Le

< |ISC)II (Mell + I-A)? ll)

<e™ |lzl|na,

where H = L7(0,¢). The inequality (ii) follows from Remark 2.2.1, Lemma 2.2.2

and the estimate

IS()zllxs < (1 + 2) |I(-A)?5(@)2lla

= (1+ 8) I-A)? Dem < 2140 > dal _ °° ny Ant nr =(1+9IID Fe < 2,¢n > V2cos =|

n=1

< (142) (sup{ ret n= 1,2,... elle

17

and

sup{-elntn ; n=1,2,...}< {2

wa

a

~e

_ 1 I 0

t

where \,, = —en?x?/f? and ¢,(z) = V2sin 2% mr,n=1,2,---. 0

REMARK 2.2.5. The inequality (2.2.9) holds for every z € H = L(0, 2), since the

semigroup 5(t) is analytic.

Now we have the well-posedness and stability properties of Burgers’ equation

(2.2.1) on the space H4(0,2). The following theorem is an application of Theorem

5.1.1 in [21].

THEOREM 2.2.6. For any given 8 > 0,0 < B < y = ex?/é’, there is a p =

p(l,€,8) > 0 such that for any initial data z9 € H}(0,2), with llzollu: < 5, there

is a unique solution z(t) = z(t,0; zo) € H}(0,) of equation (2.2.1). Moreover, the

solution satisfies the inequality

llz(t,0; zo) [laa S 2e7""|Izollnz (#2 0) (2.2.10)

and p = p(£,¢€, 8) > 0 can be chosen to satisfy

vel(y — B) <P < at O(lyGo Bt Vom) (2.2.11)

PROOF: Note that 0 is an equilibrium point for the system (2.2.3). Since D((—A)?) =

Hj, if the nonlinear term f(z) = —zz’ satisfies the hypothesis (F) in Section 2.1

18

with index p = 7 then, by Theorem 2.1.1, we have a unique local solution z(t, 0; 29)

on the space Hj. It is easy to see that

lI F (21) — F(z2)Ilz2 S (Tleallzs + lleellaa) llza — 22lln

for all z,,z2 € Hj, uniformly in t > 0. Hence, f satisfies the hypothesis (F).

For the global existence and uniqueness of the solution z(t, 0; zo) € Hj, let zo be

any initial data in Hj with ||zol|yi < 3, where p = p(£,€, ) satisfies the condition

(2.2.11). It follows that

1+2 [ 1-2) (1+ 2)x 1 —e *ds + ————_ } < -, 2.2.12

pt V2e Jo $ By? 2 \

since So. eM ds = —= Th ink = =} where [ is the Gamma function

and ['(4) = x. Let lIzollu1 < §. Then, by the local existence property, there is

a unique solution z(t,0; 20) € HZ satisfying the inequality ||z(t, 0; zo)||H2 <p on an

interval [0,t,) for some t; > 0, where t, is chosen as large as possible. We will show

that t, must be infinity. Suppose that t, is finite. Then we must have ||z(t,)|| Hi 2 p.

Note that, on the interval [0, ¢,),

IIF(2(t))lle = I — 2(¢)2"()Ilar S (Ile) Ila)? S 0”, (2.2.13)

where ’ = A Lemma 2.2.4, Remark 2.2.5 together with inequalities (2.2.12),

19

(2.2.13) yield

le(llag = [15(ta)e0+ ” S(t — )f(2(s))ds lls

<8(t)eollig + J UNS — )fCe(s))llag a - 1+é@ 1 (1+ 2)x < “ti + |

S¢ [IZol lzza p 0 { V2 Vu —s e }e M4-9) ds

p 14+2 f° 1 _ wa fe - S5t+P =e ds + eds pte tae, ys zs

p oe 7 + Ooo =—-+ ~~

PP S379 =p.

This is a contradiction. Therefore, the unique global solution z(¢, 0; z9) exists. More-

over, from the above estimate, we know that if ||zo||#: < $ then ||z(¢, 0; 20)|lz1 < p

for all t € [0, 00).

Finally, we will derive the stability result (2.2.10). Let w(t) = sup{ ||z(s)||4: e*|0<

3 <t}. We then have

llz(#) Ile

<(\IS@zollag + f IIS s)F(e(9)llagds) itt 1 4 i+ On

{Tee Feast £

a 1+2@ 1 14+ 2)x < e7 Molla vf ot af ; )

< e (e“ [Izollan + ye |Iz(s)| Fn ds )

) eX O-PE9) |I2(s)|| ye? ds

20

142 ft 1 caves L+é)r_, f° _w_ays S [lzllag + ero fo meer 9 ds) w(t) ¢ FP 9 [ere ds) w() 140 f° 1 (14 0)m

< |l2ollx2 + of Ve Sh nme ds + ty —B) } w(t)

1 < Ileollng + 51(t).

Therefore, w(t) < 2||zo||q: and ||z(¢)|lz < 2e~**||zolli- a

REMARK 2.2.7. Rankin, ({34]), considered well-posedness properties for a certain

type of semilinear evolution equations where the nonlinear terms are in divergence

form. According to his results, we can see that equation (2.2.1) has a unique (strong)

solution for initial data in L?(0,2), p > 4. To get this result, he used the analyticity

of the semigroup S(t) and the fact that the differential operator # on H}(0, 2) can

be represented by

Ve— =(—A)?B (2.2.14)

for some bounded operator B € L(H), where H = L7(0,2). In general, it is not

true that /e a = (—A)?. This result can be used to analyze the stability property

of the solution of Burgers’ equation with initial data in L?(0,2), p > 4.

Chapter III. Linear Control Problem

and Exponential Decay Rate

of the Controlled Burgers’ Equation

In this chapter we consider a distributed control problem for Burgers’ equation.

As we noted in Chapter II, the open-loop solution of Burgers’ equation decays

exponentially in the topology of an energy space (see Theorem 2.2.6). However,

the decay rate depends on the viscosity « > 0. We now explore the possibility of

obtaining an exponential decay rate independent of viscosity by feedback laws.

In Section 3.1 an abstract framework for a distributed parameter control prob-

lem will be given. The results in Section 3.1 will be applied to the Burgers’ equation

in Section 3.2. For the distributed parameter control problem we introduce two

approaches. First, after stabilizing the solution using bounded output we let the

viscosity take care of any possible “steep” gradient nature of the solution. Our

second approach is to stabilize both the solution and its gradient through an un-

bounded observation. Since for both cases the abstract frameworks are similar we

will consider the first case as a corollary of the second one.

21

22

3.1. Distributed Parameter Control Problem.

The basic model is governed by

£ a(t) = Az(t) + Bu(t), z(0) = 2 € A, t (3.1.1)

y(t) =C2(t), t20,

where U and Y are Hilbert spaces, u(-) € L?(0,00;U), y(-) € L?(0,00;Y), and

A is the infinitesimal generator of an analytic semigroup S(t) on H. Assume that

B € L(U,H) and A is self-adjoint with compact resolvent. In order to allow for

possible unboundedness of the operator C’, we assume that C' € L(W,Y), where W

is a Hilbert space such that

D(A)CWCH (3.1.2)

with continuous dense injections. Of course, the system (3.1.1) is understood in the

mild form

z(t) = S(t)zo + / S(t — s)Bu(s)ds

° (3.1.3)

y(t) = CS(t)zo + of st — s)Bu(s)ds, t>0.

REMARK 3.1.1. The output y(t) defined by equation (3.1.3) is well-defined for all

t > 0 and all z € H because the semigroup S(t) is analytic. But, y(0) = Czp is

defined only for z € W.

Since the ouput operator C’ may be unbounded, we need the following hypoth-

esis.

23

(H1). For each T > 0, there is a constant cr > 0 such that

T

[ S(t) z||2 dt < cB flel|2, (3.1.4)

for all z € W.

REMARK 3.1.2. (i) If the observation operator C is bounded on the state space H,

then W can be chosen as W = H. In this case, the hypothesis (H1) is automatically

satisfied.

(ii) If (H1) is satisfied we understand y(t) in (3.1.3) as a continuous extension

to H of the mapping z € W + y(-) € L?(0,T;Y).

We now consider the performance index

J(u) = | “{Ily(OIR-+ < u(t), Rut) >u}dt, (3.1.5)

where y(t) is given by equation (3.1.3), and R € L(U) is positive definite satisfying

the inequality < u, Ru >y > 4||u\|?, for some 7 > 0 and for every u € U. The

linear quadratic regulator problem is :

(LQR) : Find u(-) € L£?(0,00;U) minimizing the cost functional given by equation

(3.1.5) subject to the system (3.1.3).

For the existence of an admissible control u such that J(u) < oo and for the

exponential stability of the closed-loop system we need the following two hypotheses.

24

(H2). The system (3.1.1) is stabilizable in the sense that there is a feedback

operator K € £(H,U) such that the closed loop semigroup Sx(t) € £(H) given by

t

Sx(t)z = S(t)z+ | S(t — s)BKSx(s)z ds (3.1.6) 0

for all t > 0 and z € H decays exponentially.

(H3). The system (3.1.1) is detectable in the sense that there exists an operator

F € L(Y, H) such that the output injection semigroup Sr(t) € C(H) given by

t Sp(t)z = S(t)z + [ Sp(t — s)FCS(s)z ds (3.1.7)

0

for all ¢ > 0 and z € W decays exponentially.

REMARK 3.1.3. (i) If (H1) and (H2) are satisfied, then for any zo € H, there is

an admissible control u,,(-) € L*(0,00;U) such that J(uz,) < oo. This finiteness of

J(uz,) follows from the observation that, by hypotheses (H1) and (H2), there is a

T > 0, a constant cr > 0 and a feedback operator K € £(H,U) such that

T

ISe(7IIlean <1 and [ |leSe(eelRedt < lel (1.8) 0

for all z € W. We now have

/ * |[CSic(t)2||2 dt < &(—_ ) Helly (3.1.9) 0 1 — |[Sx(T) lean

Let u,,(t) = KSx(t)zo. Then it is easy to see that J(u,,) < co.

(ii) Let (H1) and (H3) be satisfied. Then for any 29 € H and u(-) € L?(0,00;U)

with J(u) < 00, z(t) defined by equation (3.1.3) is in L?(0, 00; H), ([33, p134-135)).

25

The following result which is obtained using the theory developed by Pritchard

and Salamon, ({[33, Theorems 3.3,3.4]). Since the basic arguments are similar to

those in [33] we only give a brief sketch of the proof.

THEOREM 3.1.4. Let hypotheses (H1), (H2) and (H3) be satisfied. Then there

is a unique optimal control u(-) € L?(0,00;U) for the linear quadratic regulator

problem (LQR) and u(-) is given by the feedback law

u(t)= —R'B*llz(t), t>0 (3.1.10)

where II € L(H) is the unique nonnegative self-adjoint solution of the algebraic

Riccati equation

A*llz+TAz—JIBR'B*Iz+C*Cz=0 (3.1.11)

in W* for every z € W. Moreover, the closed-loop semigroup Sp(t) € £(H) decays

exponentially.

SKETCH OF THE PROOF: The existence of a nonnegative self-adjoint solution II €

L(H) satisfying equation (3.1.11) for every z € D(A) and the uniqueness of an

optimal control u(-) given by the formula (3.1.10) follow from the hypotheses (H1),

(H2), and Remark 3.1.3.(i). On the other hand, the right hand side of the equation

NAz=—A*llz+IIBR'B*Mz-—C*Cz (3.1.12)

is well-defined for all z € W. Thus, we can extend the left hand side of equation

(3.1.12) to z € W continuously, since D(A) is densely embedded in W. Finally,

26

the uniqueness of a nonnegative self-adjoint operator II and the exponential sta-

bility property of the closed-loop semigroup Sy(t) follow from (H1), (H3), Remark

3.1.3.(ii) and work of Datko, ([13, Theorem 1]). g

COROLLARY 3.1.5 (BOUNDED OUTPUT). Let (H2) and (H3) be satisfied. If the

ouput operator C' is bounded, i.e., C € L(W,Y) and W = H, then all conclusions

of Theorem 3.1.4 hold under the condition W = H

3.2. Applications to Burgers’ Equation.

Now consider the linearization of Burgers’ equation given by

9 o(t,2) =e v(t ) t>0 O<r<e Ot ? —™ Ox? » TZ), ?

v(t, 0) = v(t, 2) =0 (3.2.1)

v(0, x) = v9(z),

where € = =, Re is the Reynolds number. Introducing z(t)(-) = v(t, -) as in Section

2.2 we can reformulate equation (3.2.1) into an abstract Cauchy problem

d He) = Az(t), 2(0) = 2% (3.2.2)

on the state space H = L?(0,), where Ad = €¢” for all 6 € D(A) = H7(0,2)N

H}(0, 2), (see Section 2.2).

For the control input operator B and the observation output operator C' we con-

sider the Hilbert spaces U = R, W = H2(0,0) = D((—A)?), and Y = R*t™,

27

Assume that B € £(U, H) and C € L(W,Y) are defined by

Bu = W(-)u, b(-) € A, uEU (3.2.3)

and

Cz = (2(21),...,2(2x), 2 (Hr), ---52 (Ym), (3.2.4)

where 7,,9; € (0,2), 1<i<gk,1<j <m, and

1 £ +6 7 1 ¥jt+6

2(z;) = af z(z)dz, z (9) = 5 / .? (x)dz. (3.2.5) 95-46

In equation (3.2.5), 6 > 0 is chosen so that (z;—6,7;+6) C (0,2) and (9; -6,9;+6) C

(0,2) forall <i<k,l<j<m.

Now consider the following linear control problem

(LCP): Find u(-) € £*(0,00;U) minimizing the performance index

J(u) = / {|ly(t)||P+ < u(t), Ru(t) >y}e?*de, a>0 (3.2.6) 0

subject to the governing equations

£ 2(t) = Az(t) + Bu(t), z(0) = 2 (3.2.7)

y(t) =Cz(t), t20,

where A, B,C are as above and R € L(V) is positive definte.

REMARK 3.2.1. (i) If the output operator C € L(W,Y) is given by

Cz = (3(%1),..., 2(Zx)) (3.2.8)

28

then C is a bounded operator on H. In this case, W and Y can be chosen as W = H

and Y = R*.

(ii) For each i, 1 <i <k, 2(%;) given by equation (3.2.5) represents an average

value of z(z) over a small neighborhood of %;. We can regard each Z;, 1 <i <k,

as the location of a “sensor”.

Similarly, y;, 1 <j <m, can be considered as the location of a sensor measuring

the average of the gradient of z(z) in the é-neighborhood of 4;.

(iii) The weight function e?* in the definition of the cost functional J will play

an important role in the exponential decay rate (see Theorems 3.2.8 and 3.2.11). It

gives rise to a question of existence of an admissible control u(-) such that J(u) < oo.

For the control problem (LCP), we introduce an “a-shifted” control system,

((18]). Let Z(t) = z(t)e, a(t) = u(t)e™ and g(t) = y(t)e%’. We then have a

modified linear control problem

(LCP)* : Find @ € L?(0,00;U) minimizing the cost functional

i(a) = [ (l@IZ+ < a(t), Ra(t) >u} dt (3.2.9) 0

subject to

£30) =(A+al) s(t) + Balt), (0) = 2 (3.2.10)

g(t) =CH(t), t>0.

Of course, the solutions for (3.2.10) are taken as mild solutions. If we solve the

29

problem (LCP)* and apply

u,(t)=e“a(t) (t2>0) (3.2.11)

to the original control system (3.2.7), then the resulting optimal trajectory Z,(t)

will satisfy the inequality

Il Za(4) || S Me™ ||zo|Lx, (3.2.12)

where M > 1 is a constant and a > 0 is the desired degree of stability.

REMARK 3.2.2. A discussion of the “a-shifted” problem for finite dimensional sys-

tems first appeared in [1]. Anderson and Moore showed that, for finite dimensional

systems, the control problem (LCP) is “equivalent” to (LCP)* in the following

senses :

(i) The minimum value of J defined by equation (3.2.6) is the same as the

minimum value of J given by equation (3.2.9).

(ii) If a(t) = g(2(t)) is the optimal control for (LCP)*~ for some function g, then

u(t) = e~%'g(z(t)e®) is the optimal control for (LCP) and conversely.

Now, we apply the results in Section 3.1 to the problem (LCP)* in order to

obtain an optimal control & € L?(0,00;U) for (LCP)”.

REMARK 3.2.3. (i) From Remark 2.2.1, the spectrum o(A+aJ) of the infinitesimal

30

generator A+ al consists of all eigenvalues Ayn, n = 1,2,---, given by

en? x? Nan = a 2. naa R (3.2.13)

and for each n, n = 1,2,---, the eigenvector ¢a,n corresponding to Ag, is given by

ban(x) = V2sin oe. (3.2.14)

(ii) We are interested in the stabilization problem for the system (3.2.1) with

small viscosity ¢ = m4 > 0, ie., high Reynolds number Re. Thus, we assume that

for any given a > 0 there is a positive eigenvalue A, n.

Let a > 0 be given and let

2 ed en’ 1 a 29}. (3.2.15) Na =max{neEN:dAgn=a-

Since A is self-adjoint (see Remark 2.2.1) and the set { dan: =1,2,---} is a basis

for H = L7(0,2), we can express W = H}(0,@) as

oo

W={z€H:) n?|<2z,$an> |? <0} (3.2.16) n=1

and identify z € H with the sequence {< z,¢un >}nen- Assume that 5, € U and

Cn € Y satisfy

Bu= {< b,,u >}nen and Cz= Soen < z, ban > (3.2.17)

n=1

and that

oo oo

1 Sr lballe <0, S>=llenll <o0, n=1 n=1

31

(see [33, p137-143]).

Now, we consider hypothesis (H1) for the a-shifted system (LCP)*.

LEMMA 3.2.4. For each T > 0, there is a constant Cr > 0 such that

T

/ ICS(t)zllp at < Ble, (3.2.18) 0

for all z € W = Hj(0,2), where H = L7(0,2), the output operator C' is defined by

equation (3.2.4) and § (t) is the analytic semigroup generated by A+ al.

PROOF: Since the infinitesimal generator A + al of § (t) is self-adjoint, the semi-

group S(t) can be represented by

Cc

S(t)z = So! < zyban >H ban n=1

for all z € H, where ¢an(z) = V2sin 2 is the eigenfunction corresponding to the

eigenvalue Agn = a —en?x?/£? of A+al,n =1,2,---. Hence, for any z€ WC H,

we have

OS(e)z IF = COS te! <2, ban >H ban) IlP n=1

oO

= || Sedan! <2, ban >H C(da,n) lly n=1

fo)

< Soe] <2, ham >H [I] C( arn) IP - n=1

By the expression (3.2.17) of the ouput operator C, it is easy to see that, for each

32

oo

C(¢a,n) = Soc < Pan) Pa,j >= Cy. (3.2.19)

j=1

Let cn = (Cniy°** »Cnjky Cnkt1)°** > Cnkt+m) € R*+™. Then it is easy to see that, by

the definition (3.2.4) of the operator C,

V2 . nxt; . nxé V2 ny; . nd

Cui = Te sin —— sin — and Cuktj = = cos —— sin — (3.2.20)

for 1 <i < kj 1 <j < m. Thus, lce,;| < vot and |cne+j| < va for all n =

1,2,---,1<2<k,1<j<™m. Therefore, for any z € W,

OS (t)z [2 < So eo] < 2, bam >H [IC (an) IF

n=l

oo k+m

= > ePant | < 2,¢an >H Kos lent”) n=1 i=1

oO

2ke? 2m 2rAa,nt 2 <yie |< 2, $an >H |"(—35 $2 n=1

2k 2m Qrant 2 =(GR tp) Le |< z,¢a,n >H | .

n=1

Note that, from definition (3.2.15) of na, Aan = a—En? x? /l? < 0 for alln > n +1.

here, without loss of generality, we assume that Ayn > 0 for all n = 1,2,--- ng.

33

Let M = (7S + 24). Then

T

[ ICS (t)z||3 at 0

00 T

<MYO( [ Pertdt)| <2,bon>P n=1

=MIS“( [ Mentdt)| <2dan> P+ Sof Peatdt) <iden> Pl N=Na+1

< MID | (e%eT _1)| <z, dan >[Pt Ss) T|<z,¢an> [7]

n=1 n=Na+t4l

1 < Mi _ (ePen? _ DY 1< aon > P+ y |< 2, $an > [7]

n=Na4l1

1

<M. (ee? — 1) +T] I lzllz,

where we have used the facts that 0 < \en, = @—en2n?/l? < don = a — en? a? /l?

for all n = 1,2,--- ng +1. By letting & = M[y2— (e¥e1* —1)+T] > 0, we

have proved the lemma. §

The following lemma is an application of stabilizability and detectability results

of Pritchard and Salamon, ([33, Section 4.2]).

LEMMA 3.2.5. For each n = 1,2,--+ ,nq, let

City 2n

={5: t=1 -,n—1} and Yn ={ :j3 =0,1,--- ,n—-1}.

(3.2.21)

Then the following statements hold.

34

(a) by =< B(-), dan ># O for all n = 1,2,--- ,n, if and only if the system

(3.2.10) is stabilizable in H.

(b) Choose 6 > 0 so small that 6 < 7 Then for each n = 1,2,--+ ,n,, there

exists at least one Z; or yj, 1 <1<k,1 <j <™m, such that

Zi ¢ Xo,ny Yj ¢ Youn

if and only if the system (3.2.10) is detectable through C' € L(W,Y).

PROOF: (a) From Remark 3.2.3, we know that the spectrum o(A+ al) of Atal

consists of all eigenvalues \,,, = a — en?x?/f?7 <0, n =1,2,---. Thus, for all n,

n> nq, +1, we know that Aun < 0. Let H, be the linear span of eigenfunctions

Pan>***,Pnq- Then the dimension of H, is n, and hence the system (3.2.10) is

stabilizable if and only if the projected system of (3.2.10) onto H,, is controllable if

and only if b, =< W(-), dan > 0 for all n, n =1,2,--- ,n9.

For (b), let en = (Cniy*°* 5 Cnjky Cnktiy°** > Cnktm), 2 =1,2,--- ,nq, be defined

by equation (3.2.19). Then, by equation (3.2.20), we have

J20 NNz; n7wd J2 NTy; nré ;= i = — in —— 2.2 Cn,i = sin 7 sin ? and Cnrk4j 5 COS sin 7 (3.2.22)

for 1 <i < k,1 <j <m. By the dual statements of (a), (3.2.10) is detectable

through C € L(W,Y) if and only if c, # 0 for all n = 1,2,---,n,. Hence, (b)

holds. y

REMARK 3.2.6. Ifn, = 1, then X,,1 is the empty set.

39

We now return to the original control problem (LCP). The following theorem is

the main result for our control problem (LCP).

THEOREM 3.2.7. Let a > 0 be given. Suppose that b(-) € H = L?(0,2), 6 >

0,7,,1<%1< k, and y;,1 <j <M, satisfy the conditions (a) and (b) of Lemma

3.2.5. Then there is a unique optimal control u,(-) € L?(0,00o;R) for the problem

(LCP) such that

i.(t)= —R"*B*T,Z(t), t2>0 (3.2.23)

where Z,(t) is the corresponding optimal trajectory and Il, € £(H) is the unique

nonnegative self-adjoint operator satisfying the algebraic Riccati equation

(A + a1)*T1,z + II,(A + al)z — T,BR'B*I,z + C*Cz =0 (3.2.24)

in W* = H-1(0,2) for every z € W = H}(0, 2). Moreover, the closed loop semigroup

Sn, (t) € £L(H) satisfies the following stability property

lI Sta (4) Ica) < Melt}! (3.2.25)

for some constants M = M(a,e¢) > 1 and w =wu/(a,€) > 0.

PROOF: By Lemmas 3.2.4 and 3.2.5, we know that the a-shifted control system

(3.2.10) satisfies all hypotheses (H1), (H2) and (H3) in Section 3.1 (with z(t), y(t), u(t),

A, S(t) and J replaced by Z(t), 9(t), u(t), Ata/, S(t) and J, respectively ). Hence,

by Theorem 3.1.4, there is a unique optimal control &(t) for (LCP)* and the corre-

36

sponding closed-loop semigroup 5 (t) decays exponentially, i.e.,

US()leun < Me = t > 0 (3.2.26)

for some constants M = M(a,e) > 1 and w =w(a,€) > 0. Moreover, @(t) is given

by

a(t) = —R7' Bll, 2(t) (3.2.27)

where 2(t) is the resulting optimal trajectory for the a-shifted system (3.2.10) and

II, is the unique nonnegative self-adjoint solution of equation (3.2.24). Since the

semigroup S$ (t) is generated by A+ al — BR™'B’*II,, the infinitesimal generator of

the closed-loop semigroup Sy, (t) for the original system (3.2.7) is A— BR B*Iq.

Hence, Sy,(t) = S(t)e~* and, by the relation (3.2.26), Sy,(t) satisfies the in-

equality (3.2.25) with M = M. Moreover, the optimal control t,(t) for (LCP)

is given by the formula (3.2.23), since t,(t) = u(t)e"* = —R-'B*I,2(t)e"™ =

—R-'B*T1,Z,(t), where Z,(t) = 2(t)e~* is the corresponding optimal trajectory for

the original system (3.2.7). This completes the proof. §f

The optimal control a@,(-) € L?(0,00;R) obtained in Theorem 3.2.7 is given by

the feedback law (3.2.23). Define the feedback operator K, € £(H,U) by

K, = —R'B' ly. (3.2.28)

Then the optimal control u(t), t > 0, is given by

Ua(t) = Ky Z(t) (3.2.29)

37

and the infinitesimal generator for the closed-loop semigroup Sj, (¢) is

A+ BK, = A—BR'B'Tly. (3.2.30)

Recall that H = L*(0,2) and U = R. Thus, by Riesz representation theorem,

(see, e.g., [11, p13]), there is a unique feedback gain function ka(-) € L?(0,€) such

that

t Kaz = / ka.(s)z(s) ds (3.2.31)

0

for all z € L(0, 2).

We are now at a point where we can obtain a stability result for the controlled

heat equation.

COROLLARY 3.2.8. Let a > 0 be given. Suppose that b(-) € L°(0,2), 6 > 0,

Z,1<i<k, andy;,1 <j <™m, satisfy the conditions (a) and (b) of Lemma 3.2.5.

Let the feedback gain function ka(-) € L?(0,@) be given by the formula (3.2.31).

Then for any initial data z € L?(0,0), there is a unique (strong) solution of the

controlled heat equation

0 o b k d —2z(t,rz) =e—>2(t a(s)2(t, Fiith2) = ex ae(ta) + H(z) | ko(s)e(t,s) ds

2(t,0) = z(t, £) = 0 (3.2.32)

2(0, xz) = zo(z) € L7(0, 2)

and the solution z(t)(-) = z(t,-) satisfies the following inequality

I 2(2)() Ilne0.y < Mae" @*™ [z0llz2(0,9 (3.2.33)

38

for some constants M, = M,(a,¢) > 1 and w =wu/(a,«) > 0.

PROOF: The existence and uniqueness of a strong solution z(t)(-) = z(¢,-) with

initial data zp € L?(0, 2) follow from the fact that the closed-loop operator A+ BK,

defined by equation (3.2.30) generates an analytic semigroup Sy, (t). The stability

result (3.2.33) follows from the inequality (3.2.25).

REMARK 3.2.9. From Remark 2.2.1, the open-loop solution z(t)(-) = z(t,-) for

heat equation (3.2.32), with k, = 0, satisfies the stability result

II 2(t) Ilz2oe) < e/ |Izollz20,2) (3.2.34)

for all z € L*(0, 2).

Now we are ready to state the main result of this section.

THEOREM 3.2.10. Let a > 0 be given. Suppose that b(-) € H = L7(0,2£), 6 >

0,2;,1<i%<k, and y;,1< 3 <™m, satisfy the conditions (a) and (b) of Lemma

3.2.6. Let ka(-) € H be the linear feedback gain function defined by the formula

(3.2.31). Then there exist constants p = p(a,¢€) > 0 and M = M(a,e) > 1 such

that for any initial data vo(-) € H9(0,¢), with ||vollaa < ghz, the controlled Burgers’

39

equation

2 tt r)=€ SF ws x) — v(t 2) v(t rt) + u(2) fs (s)v(t, s) ds ot 3 Ox? 3 ’ Oz 7 0 a 3

v(t,0) = v(t, 2) = 0, (3.2.35)

v(0,z) = v9(z) € H4(0, 2)

has a unique (strong) solution and the solution v(t)(-) = v(t, -) satisfies the following

stability property

Il o(4) Iles S 2Me~* |Joo(-)I La (3.2.36)

PROOF: Let the operators A, B, C and K, be as in this section, i.e., defined by

equations (3.2.2), (3.2.3), (3.2.4) and (3.2.28), respectively. Define the nonlinear

function f : Hj(0,£) — L7(0,2) by

f(z) = BKaz— z2' (3.2.37)

i where ' = 4. Then, the map f satisfies the hypothesis (F) in Section 2.1, since for

any 21,22 € Hi (0, 2),

I] F(z1) — F (22) IIz2(0) S (| BKallecuy + |l2illza + Ilz2lla2 ) M21 — 22llng- (3.2.38)

Note that the operator BK, is bounded on the state space H = L?(0,¢). Thus, by

Theorem 2.1.1, we have a unique local (strong) solution of equation (3.2.35).

Let Sx, (t), t > 0, be the analytic semigroup on H generated by the operator

A+BK,. Then, by Corollary 3.2.8, Sx, (t) satisfies the inequality

IS. (t) z|| S Ma et Iz || (3.2.39)

40

for all z € H and for some constants M, = M,(a,¢) > 1 and w = w(a,e) > 0.

Since Sx,(t) is analytic, for any 8 with a < B < a+vuw, there is a constant

M, = M,(a, €, 8) > 1 such that

Il Sa (t) 2llna S Mae |lz|l a3 (3.2.40)

Sia (€) lag < Mae Ulll (3.2.41)

for all z € H}. Let

M = max{ M,, M,} (3.2.42)

and choose p > 0 with

VB -—a . 3.2.43

0<?< 0 EM (3.2.43)

Then it is easy to see that

pm [- J .-(6-a)» ds < P (3.2.44) 0 vs 2

Thus, by arguments similar to those in the proof of Theorem 2.2.6 together with

inequalities (3.2.40)-(3.2.44) and the expression

v(t) = Sx, (t)vo + [ Sk,(t — s)g(v(s))ds, (3.2.45)

the unique global solution v(¢)(-) = v(t,-) for the controlled Burgers’ equation

(3.2.35) exists and satisfies the inequality (3.2.36), where g(v(t)) = —v(t)u’(t). 5

Chapter IV. Approximation and Numerical Results

In Chapter III, we considered a linear control problem. Under certain conditions

on the input and the output operators there is a unique optimal control which

guarantees the desired degree of stability for the closed-loop system (see Lemma

3.2.5, Theorem 3.2.7, Corollary 3.2.8, and Theorem 3.2.10).

In this chapter we consider various approximation schemes for the control prob-

lems considered in Chapter III. Abstract approximation schemes and its applications

will be discussed in Section 4.1. In Section 4.2, three feedback schemes are intro-

duced. Numerical results will be shown in Section 4.3. Finally, the conclusion of

this paper will be given in Section 4.4.

Throughout this chapter, we use superscript N in the designation of subspaces,

operators and matrices in the N-th approximating system and corresponding control

problem, like H%,A%,B%, etc. Hence the superscript N indicates the order of

approximation.

4.1. Approximation Schemes for Linear Regulator Problem.

In this section we consider an approximation scheme for the abstract linear

regulator problem with bounded input and output operators and then apply the

scheme to our control problem. The following approximation scheme is based on the

results from [4,23]. For more approximation schemes for linear regulator problems,

41

42

see [2,3,17] and refences given there.

We suppose that H, U and Y are separable Hilbert spaces, that A : D(A) C

H — H is the infinitesimal generator of a Co-semigroup S(t) on H and that

Be L(U,H) and CeCL(H,Y). (4.1.1)

Consider a control problem given by

d He) = Az(t) + Bu(t), 2(0)=z2€E€H

(4.1.2)

y(t) = C2(t)

and an associated performance index

Heavu) = f° {ly(tie+ < Bult), u(t) >v}at (4.1.3) 0

where R € L(U) is self-adjoint and strictly positive. We interprete the control

system (4.1.2) in the mild sense:

z(t) = S(t)z + | S(t — s)Bu(s)ds 4.1.4)

y(t) = C2(t)

The abstract linear optimal regulator problem is stated as

(R) : Minimize J(zo, u) over u € L?(0,00;U) subject to the system (4.1.4).

The following theorem is well-known, (see [4,16,43]), and can be viewed as a

revised version of Theorem 3.1.4.

43

THEOREM 4.1.1. Assume that (A,B) is stabilizable and (A,C) is detectable.

Then there is a unique nonnegative self-adjoint solution II of the algebraic Riccati

equation

A*ll + 1A — ILBR™' Bll + 0"C =0 (4.1.5)

and the unique optimal control u(-) € L?(0,00;U) is given by

u(t) = —R'B*ISp(t)z20, (4.1.6)

where Sy(t) is the Co-semigroup generated by A— BR™'B*II. Moreover, Sy(t) is

exponentially stable.

We next formulate a sequence of approximate regulator problems and present

a convergence result for the corresponding Riccati operators.

Let HN, N = 1,2,---, be a sequence of finite dimensional linear subspaces of

H and P% : H — H™ be the canonical orthogonal projections. Assume that S(t)

is a sequence of Co-semigroups on H™ with infinitesimal generators AN € L(H¥).

Let BN € C(U, H®) and CN € L(HN,Y). We then consider the family of regulator

problems:

(R) : Minimize JN (2z/’,u) over u € L?(0,00;U) subject to the control system

z(t) = SN(t)2N + [ S(t — s)BNu(s)ds, 2% (0) = 2¥ = PX x 0 (4.1.7)

y(t) = CN ZN(t), (4.1.8)

44

where

IN (Nu) = | “ {lly ()IR-+ < Ru(t), u(t) >u}ae. (4.1.9)

REMARK 4.1.2. If, for each N, (A%, BY”) is stabilizable and (A, C’) is detectable,

then, by Theorem 4.1.1, there is a unique optimal control w%(t) for the finite di-

mensional problem (R%) and it is given by

aN (t) = —R71(BN)*TIN SH (t)2N, (4.1.10)

where S/V’(t) is the Co-semigroup on H™ generated by AY — BN R-1(B%)*IIN and

IY € L(H*) is the unique nonnegative self-adjoint solution of

(AN )*T1N + TEN AN — TIN BN R-1(BYN)*TIN + (C%)*C% = 0. (4.1.11)

For the finite dimensional approximation systems, it is not clear that (A’, BY)

is stabilizable even if the original system (A,B) is stabilizable. Similarly, it is

not clear that the detectability property of (A,C’) is preserved under the finite

dimensional projections. Another question we have to consider is the convergence

of approximates IIY and a(t) to the infinite dimensional solutions II and u(t),

respectively. For these reasons, we need the following assumptions.

Let S%(t) = e4"'*, t > 0.

(A1)* : For each z € H we have

(i) SN(t)P%z —> S(t)z, and

45

(ii) SN(t)* Pz — S(t)*z,

where the convergences are uniform in ¢ on bounded subsets of [0, 00).

(A2)% : (i) For each u € U, BNu —> Bu and for each z € H, (B%)*PNz — B*z.

(ii) For each z € H, CN PN z — Cz and for each y € Y, (C’)*y —> Cry.

(A3)% : (i) The family of the pairs (AN, BY) is uniformly stabilizable, i.e., there

exists a sequence of operators KN € £(H%,U) such that sup ||K || < oo and

t>0, (4.1.12) )

NIeN _ \Je(A* +8 K # PN can < Me wyt

for some positive constants M, > 1 and w, > 0 which are independent of NV.

(ii) The family of the pairs (AY,C™) is uniformly detectable, i.e., there exists

a sequence of operators F% € £(Y, H%) such that sup ||F|| < oo and

Jaren Ot PN oy < Moe", = t > 0,~ (4.1.13)

for some constants M, > 1 and w2 > 0 which are independent of N.

REMARK 4.1.3. (i) The condition (A3)‘(ii) is a relaxation of the coercivity as-

sumption in [4], (see [23, p3]).

(ii) Suppose that BY = P’B and CN = CPN. Then (A2)* holds, since it

follows from (A1)% that PNz — z for all z € H.

By simple modification of results from [23, Theorem 2.1] and [4, Theorem 2.2],

we have the following fundamental convergence results.

46

THEOREM 4.1.4. Let (A,B) be stabilizable and (A,C’) be detectable. Suppose

that (A1)* —(A3)* are satisfied. Then, for each N, the finite dimensional algebraic

Riccati Equation (4.1.11) admits a unique nonnegative self-adjoint solution TIN such

that

sup{||IV" ||cuzx) : N = 1,2,---} < 00 (4.1.14)

and

TY PN z — Iz (4.1.15)

for every z € H. Moreover, there exist positive constants M3 > 1 and w3 (indepen-

dent of N) such that

[Je AN BER BNI PN any < M3e7“**, t>0. (4.1.16)

Next, we will apply the previous approximation results to our control problem

for the linearized Burgers’ equation. Throughout the rest of this section we assume

that R = I and @ = 1 for convenience. The governing equation for our control

problem is given by

0 0 Bi 7hb 2) =€ 9g 2(t =) + B(z)u(t), t>0, O0<2<1

2(t,0) = 2(t,1) =0 (4.1.17)

z(0, x) = zo(z),

where 6(-) € L£7(0,1). Let H = L7(0,1), Y = R* and U =R. Let Az = ez” for

47

z € D(A) = H2(0,1)N Hi(0,1) and BE L(U, H), C € L(H,Y) be defined by

Bu=W(-)u, ueU (4.1.18)

C(z) = (2(21), 2(22),-« , (Ex), 2 CH (4.1.19)

Z,+6 where 2(Z;) = + 2,6 2($)ds, Z;’s denote the location of the sensors (see Remark

3.2.1), and 6 > 0 is chosen small enough that (z; — 6,z; + 6) C (0,1) for all z,

1<i<k. Then our linear regulator problem can be stated as

(Ra): Minimize J,(zo, Ua) over Ua(-) € L?(0,00;U) subject to

Zq(t) = Sy(t)zo + [ S(t — s)Bu,(s)ds, (4.1.20)

Ya(t) = Cz,(t), (4.1.21)

where a > 0, z(t) = z(t)e™, y(t) = y(t)e™, ua(t) = u(tje™, S(t) is the analytic

semigroup generated by A+ al, and

Ja(205 Ua) = | {Ilye(t)|[2: + [fua(t)]I2 }at. (4.1.22)

Assume that (A+ a/J, B) is stabilizable and that (A+aJ,C) is detectable. The

necessary and sufficient conditions to stabilizability and detectability can be found

in Lemma 3.2.5. Then, from Theorem 4.1.1, the corresponding Riccati equation and

the optimal control a,(t) for problem (R,) are given by

(A+ al)"Il, + Ha(A + al) — a BB*la + C*C =0 (4.1.23)

48

and

ta(t) = —B* Z(t), (4.1.24)

where Z,(t) is the corresponding closed loop trajectory.

For the uniform stabilizability and detectability assumptions (A3)% we intro-

duce a sesquilinear form a(-,-): V x V — C defined by

1 a(z,w) = [ ez'(x)w'(x)dz, zwevVv (4.1.25)

0

where V = H}(0,1). Note that, to allow the use of the theory of sectorial operators

and sesquilinear forms in discussing the spectra of various operators, we assume in

defining a(-,-) that the functions in V are complex valued. It is easy to see that the

sesquilinear form a(-,-) is V-coercive, ([42, p274]), i.e.,

la(z,w)| <e|lz|lv ||wlly (continuity) (4.1.26)

and

Rea(z,z) + y||z||% > ellz||% (Garding’s inequality) (4.1.27)

for all z,w € V and y > € > 0. Furthermore, it follows from the bounds (4.1.26)

and (4.1.27) that there exists, in a unique manner, an operator A € L(V,V*) such

that

a(z,w) =< —Az,w >y-y (4.1.28)

and

a(z,w) =< —A*w,z >yey (4.1.29)

49

for all z,w € V, (see, e.g., [42, pp271-275]).

REMARK 4.1.5. (i) Since Az = ez” for all z € D(A) = H? NM Hi and D(A) is dense

in V, the operator A is nothing but the unique extension of A from D(A) to V. For

this reason, we will denote A by A. Hence, throughout this section, the operator A

will be understood from the context.

(ii) Since A = A", it is easy to see that A = A*.

REMARK 4.1.6. From Garding’s inequality (4.1.27), we know that for any A € o( A),

Rer<-y<~-e. (4.1.30)

Turning next to approximations for (R,.), we divide the unit interval [0,1] into

N +1 equal subintervals to get [z;, 2:41], 2; = Wei? i= 0,1,---,N. For each 2,

1<i<QN, let h(x) denote the linear spline basis function defined by

—(N +1)(2@— igi), 2520 < ti41 (4.1.31)

(N + 1)(z — 2;-1), @;-1<2< 2;

AN (2) = 0, otherwise.

Let H™ be the N-dimensional finite element space given by

N

HN ={)o2hN(2): 4 € R,t=1,2,--- ,N}. (4.1.32) t=1

Then we have a sequence of finite dimensional (real) subspaces HN C V, n =

1,2,---. Moreover, it is well-known, ((36],[23, p15]), that the family of H™ satisfies

the following approximation condition: ,

50

(APP) : For each z € V, there exists an element 2% € H™ such that

||z — z* |v < e(N), where e(N) — oo.

Let P: H — H% be the canonical orthogonal projection onto H™. Then, from

the approximation property (APP), it is a trivial matter to see that

PNz—+z asN-oo, forzeH. (4.1.33)

For the finite demensional regulator problem (R%) we choose

BN=PNB and CN=CPN%. (4.1.34)

Then conditions (A2)‘(i),(ii) follow from Remark 4.1.3.

To obtain the finite dimensional representation A of A, consider the restriction

of the sesquilinear form a(-,-) to HN x HN. We then have a representation A% of

A satisfying

a(z,w) =< —A%z,w > (4.1.35)

and

a(z,w) =< —(A™%)*w,z > (4.1.36)

for all z,w € HN. Equation (4.1.36) follows from the fact that H is a real Hilbert

space. We know also that AN = (A)*, since A = A”.

REMARK 4.1.7. Since H™ C H, by equations (4.1.35) and (4.1.36), it is easy to

51

see that for any 4 € o(A¥),

RerX<-y< -«€. (4.1.37)

Let S‘(t) be the Co-semigroup generated by AN. Then the conditions (A1)%(i),(ii)

follow from the results of Banks and Kunish, ([4, Lemma 3.2]). Note that S%(t) =

(S%(t))*. Moreover, for each fixed time t > 0, the convergence rates follow from the

next theorem, ([15, Theorem 4.1]).

THEOREM 4.1.8. We have the following estimates:

. 1 1 (1) [IS(@)2z- S*@)PNallv <e(F— GIF, (> 0) (4.1.38)

ee om 1 2 1 (Hi) [|S@)z- S™()P%zlla < e(y pi) (piiiella (> 9) (4.1.39)

for some positive constant ¢ which is independent of N and t.

For the condition (A3)‘(i) we need a certain preservation of exponential stabi-

lizability under approximation, (see (POES) in [4]). The following result is taken

from [4, Lemma 3.3].

THEOREM 4.1.9. Let (A,B) be (exponentially) stabilizable. Suppose that the

approximation condition (APP) holds. Then the approximations defined through

equations (4.1.34)-(4.1.36) satisfy the condition (A3)‘ (i), i.e., the family of pairs

(A%, BY) is uniformly stabilizable.

By the dual arguments of Theorem 4.1.9 we can see that the condition (A3)% (ii)

52

holds under the assumption that (A, C’) is detectable. We summarize our discussion

up to this point as the following theorem.

THEOREM 4.1.10. Let (A+al, B) be stabilizable and (A+alI,C) be detectable.

Let AN, BN, CN be defined as in equations (4.1.34) and (4.1.35). Then we have

WY pP%z—Tl,z, zeH (4.1.40)

and

SN(t)P%z — S(t)z, ze€H (4.1.41)

where the convergence is uniform in t on bounded subsets of [0,00), P% is the

orthogonal projection onto H™, and II% satisfies

(AN + aN) TIN + TIN (AN + aI) — TY BN(BY)*TIY + (CX)"CN = 0. (4.1.42)

REMARK 4.1.11. Note that S(t) = S(é)e~*t, SN(t) = S%(t)e7*t, where S(2),

SN (t), S(t) and S(t) are semigroups generated by A, AY, A+ al and AN + al,

respectively.

Next, consider the matrix representations of operators on the space H™. Let

the approximate solution z¥(t, x) of z(t, 2) on H™ be given by

N

2N(t,2) = > zi (t)hN (zx) (4.1.43) t=1

for some z(t) € R, i = 1,---,N, where 2(t,z) is the open loop solution (with

u = 0) for equation (4.1.17). Then, from equations (4.1.34) and (4.1.35), we have a

53

finite dimensional ODE system

a" Ste") = ANN} + BUC), (4.1.44)

where {z(t)} = [2i" (2), vs zh (t)]",

[GX] = [< hn AN >]NxN

[4 1 0 O ... O 1 4 1 0 ... O

1 01 4 1 .. O

a rr 4.1.45

even}, 0 ef 7 0 1 4 | 0 0 1 4) nyn

T—2 1 0 0 07 1 —2 1 0 0 0 1 —2 1 0

[A”] = e(N +1) te oo (4.1.46)

0 1 -—2 1 LL 0 coe 0 ] —2 | NxN

{BY} =[<b, kV >, < bhY >,---,< BAN >’, (4.1.47)

where < 6, Ay >= fo b(x)hN (x)dz, 1<j< WN. Since [GY] is invertible, by multi-

plying [G*]-! to both sides of (4.1.44), we get

£(2%(t)) = [AN] {2%(t)} + {B™ u(t), (4.1.48)

{2% (0)} = {20},

where

[AX] =[G*]}"[A™], {BN} = [G*){B"} (4.1.49)

and {24’} = [G%]“[< zo, hl >,--- ,< 2, h® >I’.

54

Next, consider a representation C™ of the operator C on H%. It is easy to see

that CY : HN — R* is given by

[CN] = [AN (z,)laxn, | (4.1.50)

where AN(z;) = 2, fr? aN(a)dz,1<Si<k, 1 <j<QN.

Finally, we have a finite dimensional Riccati equation

(AN + @IN)*TIN + TN(AN + aN) — TN BN(BNY*TIN 4+ (CX)*CN =0 (4.1.51)

and the corresponding feedback gain operator K* given by

KN = —(BY)*1. (4.1.52)

Therefore, the closed loop system (4.1.18) can be represented by

£ (2% (0) = (AN + BYK®)(2"(0)} t (4.1.53)

{z™(0)} = {z9'}.

REMARK 4.1.12. Consider an unbounded observation operator C € £(H2(0, 1); R*t™)

given by equation (3.2.4) in Section 3.2, i.e.,

Cz = (2(21),-++ , 2(Zk), Z (91), -- «9 Z(Gm))s (4.1.54)

where 2(Z;) = 3 ait 2(z)dz, 1<ic<k, 2'(y;) = Hp 2! (2)de, 1<j<™m, are

defined by equation (3.2.5). Then the finite dimensional approximation C% of C on

H™® can be represented by

~ YN oa & ), (4.1.55)

55

where CN: HN — R‘ and CN: HN — R™ are given by

[CPX] =[(C® )ijlexw, (CN) = AN(z:), (lSiskl<j<N) (4.1.56)

[CP] = (CM )islmxn, (CH )ig =(N +10" (G:), (LSism1l<j<N) (4.1.57)

where hY, 1 <j < N, is the linear spline function given by equation (4.1.31) and

py, 1<j<N, is defined by

l, Tj-1 < 2< 7;

wN(z)= 4-1, 2) <2< thy) (4.1.58) 0, otherwise.

Thus [C%] is a (k-+m) x N matrix, and the corresponding finite dimensional Riccati

equation and feedback gain operator KN are given by equations (4.1.51) and (4.1.52)

with CN = CN.

Finally, we discuss an algorithm for finding the unique nonnegative self-adjoint

Riccati solution for equation (4.1.51). We employ the Potter’s method, (see, e.g.,

[35]), to obtain II. The first step in Potter’s method is to form 2N x 2N matrix

MN = ete. - Ue r om") . (4.1.59)

Next, find all eigenvalues and eigenvectors of M™ and form the matrix

N

ZN = 3 | . (4.1.60)

where the columns of Z™ are the eigenvectors of M™ corresponding to the eigen-

values with positive real part. When eigenvalues occur in complex conjugate pairs,

56

so do the eigenvectors. In this case, the real and imaginary part of the eigenvector

each forms a column of Z%. Finally, the solution to the Riccati equation (4.1.51) is

given by the formula TIN = QN(Q)’)-1.

REMARK 4.1.13. From the numerical results we found that the Riccati solution

operators IIN = II (e) blow up when the viscosity « > 0 goes to 0 for fixed a > 0.

Also, when a goes to infinity with € fixed the same phenomenon has been observed.

4.2. Feedback Schemes.

From Sections 3.1, 3.2 and 4.1, we know that the optimal control u(-) € L?(0, co; UV)

for our linear regulator problem is given by a feedback form

u(t) = K2(t) (4.2.1)

and the feedback operator K € £(H,U) can be represented by

1 K2(t) = / k(s)z(t, s)ds (4.2.2)

0

for some k(-) € H, where H = L7(0,1), U = R, z(t)(-) = z(t,-) is the corresponding

optimal trajectory, (see Theorems 3.1.4, 3.2.7 and 4.1.1). Therefore, the infinite

dimensional closed loop system for Burgers’ equation (3.2.35) is given by

F a(t) = (A+ BK)z(t) + f(z(t)), t>0 (4.2.3)

2(0) = z

57

on H = L*(0,1), where Az = ez” for z € D(A) = H?(0,1)NH4(0,1), Be L(U, H),

Bu = b(-)u for u € U, b(-) € H, and f(z) = —zz’, ' = 4, and z= € H or Hi

depending on the observation operator C.

In this section we introduce three schemes for feedback law; feedback by the

optimal L?-gain function k(-) in equation (4.2.2), the step function averaging k(-) on

each interval J; = (9;-1, y;), and by the average value of k(-) in a small neighborhood

of each center point y; of I;, where 99 = 0, ¥m41 = 1, 9;’8, 1 < 7 < m, are defined

as in equation (3.2.4).

Let the finite element space H™ be defined by equation (4.1.32) and the ap-

proximations z‘(t, 2), KN and kN(z) on H% for z(t,x), K and k(z), respectively,

be given by

N N

2N(t,2) = So zN(t)AN (a), RN (z) =D RN A(z) (4.2.4) t=1 t=1

and

KN = [oh aga IN - (4.2.5)

Then the coefficients kN’s in equation (4.2.4) are determined by the formula

[ey ky, my kn) = [e™]" 72 “Te ee (4.2.6)

where [G] is as in equation (4.1.45). Hence, on H%, the approximate control u(t)

becomes

uN(t) = KN2ZN(t) = S aN 2M (t) (4.2.7) i=1

58

and the finite dimensional system for the closed-loop system (4.2.3) is

£{2N(6)} = ((A] + {BYHA® DEN} + ML2M(OY, > 0 dt (4.2.8)

{z"(0)} = {20},

where {z%(t)} = [zN(t),--- ,zN(t)]?, [A], {B%}, {227} are defined as in equations

(4.1.44) - (4.1.49), [AK] is as in equation (4.2.6), and

FN({ZN (DY) = [GATT (2), (4.2.9)

2 (t)29 (t) + (29 (t))? —(23%(t))? — (20° (t) C22") + (22"(t))(23' (2) + (23°)?

—(2K_2(t))? - (=H_o(2))(R_a(¢)) + (zN-1(t))(zn(t)) + (zn (t))? —(2N—1(t))? — (zn_-1(¢))(2n (2)

To solve the nonlinear ODE system (4.2.8) we use the 4-th order Runge-Kutta

P(e") =

method, (see, e.g., [38]).

REMARK 4.2.1. For the representation (4.2.5) of K%, we used the fact that the

input operator B € L(U, H) is 1-dimensional.

Now, consider the following approximation schemes for various feedback laws.

Scheme 1. First, we consider a feedback form (4.2.2), say K,, corresponding to

the optimal control u(t). Then, on H%, KX is determined by

KN = 72's Whi x

= ~(B"}" I

= —[by’, bf, “ee , ON [TT] (4.2.10)

59

and hence

N

ul (t) = KN2N(t) = Sah), (4.2.11)

where bN =< 0(-),hAN >y, 1 < i < N, are as in equation (4.1.47) and [II] is

the Riccati matrix solution for equation (4.1.42). The subscript o indicates the

“optimal” feedback.

REMARK 4.2.2. For the control problem with bounded observation operator we

use only Scheme 1. Numerical results are presented in Figures 4.3.2-4.3.9. Scheme

2 and Scheme 3 are designed for the problem with unbounded output operators. Of

course, those schemes can be applied to the first case with intervals [Z;_1,Z;] and

center points Z;,1<1< k.

Scheme 2. Consider the following feedback form K,:

zy mti

K,2z =| (s° a;X1,(s))z(t, s)ds, (4.2.12) j=l

where Yo = 0, ¥m4i1 = 1 and for 7 = 1,--- ,m+1,

aod a; = — k(s)ds

’ Yi — Yj-1 JI, (s)

= the average of k(x) over the interval J; = (¥;-1, 9;),

_ l, «re I;

XU) = 0, otherwise.

and for each 7, 1 <j <m, 4; is the location of sensor measuring 2 z(t, 2). Then

60

on H%, the control function uN (¢) becomes

ua (t) = Kz" (2) 1 m+l

=f (S 5 aN x1,(s))z"(t, s)ds j=l

= So uy u 2N(t), (4.2.13) s=1

where

m+1

= 2% ‘| hN(2)de , a =—_— sae | hN(x)dz). (4.2.14) Yj — Yj-1

Thus the feedback operator KN is represented by

Ky’ = [uy uy’, a) un]. (4.2.15)

Scheme 3. Consider the feedback form

m+1

Kapz = » a;2(t, 9;), (4.2.16) j=l

where a,;’s are as in Scheme 2 and for each 7, 1 <j <m, 4; is the center point of

the interval J; = (9;-1,9;) and 2(t,9;) = 3; Is Specs 2 5; 2(t,s)ds, 0< 6 << 1. It follows

that on HY’,

9;+6 N

Mbi=sg [oa Onl Yj-§ gay

= » lee l AN (s)ds), (4.2.17)

w=l

61

and

u(t) = KNZN(t)

m+1

= >) af z(t, 95) j=l

N N m+1 N 1 §;+6 N

=O Lag [ [AN (s)ds) t= j=1 J

= NN N wae nl [Bt on =Serey, Wa Pare fo a(o)as). 2, 2 "26 Ji;-5 (4.2.18)

Hence, the feedback operator K pl is given by

KN = [oy vi’, vy). (4.2.19)

4.3. Numerical Results.

In this section, we discuss how our results work for relaxation of “steep” gradient

of the solution for Burgers’ equation, with Dirichlet boundary condition, through

numerical experiments. The computer codes described in this section were imple-

mented on a VAX 8800.

For Examples 4.3.1-4.3.3, the length @ for space domain, Reynolds nummber

Re, initial function zo(-) € H4(0,1) and control input function 6(-) € £*(0,1) will

be chosen as 1, 60, sin wz and e”, respectively. Thus the governing equation is given

62

a 1 2 a -f- By 7lts r)= 60 qr tts zr) — 2(t, t)ao2(t, z)+e [ k.(s)z(t, ) ds

2(t,0) = z(t,1) =0 (4.3.1)

2(0,z) = sinzz,

where the feedback gain function k,(-) € L?(0,1) will be determined by the desired

degree a > 0 of stability and the action of output operator C.

The robustness of the feedback controller exhibited, for example, in Figure 4.3.2

will be discussed in Example 4.3.4. For this particular example, Reynolds numbers

60, 80, 100 and 120 are chosen.

REMARK 4.3.1. (i) From the numerical experiments, we found that if Reynolds

number Re is less than 60, then the diffusion phenomena dominate convection phe-

nomena. In this case, the formation of steep gradient due to convection term

—2(t,x)22(t, 2) of the open-loop solution, i.e., ka{-) = 0 in equation (4.3.1), is

not clear. But, for Reynolds number greater than 60, the open-loop solution creates

“sharp” gradient in finite time (see Figure 4.3.1). Of course, the solution dies out

eventually, because of the diffusion term 1 Set, x).

(ii) The control input function b(z) = e7 is defined for all z € [0,1]. Thus,

the feedback control acts on the whole domain [0,1]. But, one can choose any L?-

function b(-) € L?(0,1) satisfying the stabilizability condition in Lemma 3.2.5. In

fact, b(z) = e” satisfies the stabilizability condition for any desired degree of stability

63

a > 0, since the coefficients 6,, n = 1,2,..., representing the input function b(-)

are not zero, 1.e., b, =< b(-), sin tz >220,1)= fo e*sinaz dr # 0 forall n=1,2,...

(see Lemma 3.2.5).

(iii) The initial function zo(z) = sin rz is chosen for our numerical experiments.

Other typical H}-functions such as the “hat function” defined by

_ J 22, z € [0, 5] zo(r) = 0n42, ze (2, 1 (4.3.2)

can be used for initial data. But, we found that the solution of Burgers’ equation

(4.3.1) with initial data zo(x) replaced by the hat function has almost similar phe-

nomena, such as the creation or relaxation of steep gradient, as those of solution

with initial data zo(r) = sin rz.

To show trajectories of open-loop and closed-loop solutions, the order N of ap-

proximation is chosen as N = 32 for both cases. And the corresponding trajectories

from time t = 0.0 to t = 1.0 are shown. The convergence of the feedback L?-gain

functions k,(-) € L?(0,1) are shown for N = 8, 16,32, 64 and 128.

EXAMPLE 4.3.1. (Bounded Observation)

The observation operator C € £(L7(0,1),R*) for this example is given by

C(z) = (2(0.3), (0.5), 2(0.75)), (4.3.3)

where 2(Z) is the average value of z(-) € L?(0,1) in a small neighborhood of @, z =

64

0.3,0.5,0.75, and defined by equation (3.2.5)

1 +6

26 Jes 2(Z) = z(s) ds. (4.3.4)

Here, 6 > 0 is chosen so small that each open interval (= — 6,Z + 6) is contained

in the whole domain (0,1). The desired degree a of stability are chosen 0.3 and

0.6 for Figures 4.3.3 and 4.3.5, respectively. For both cases, ng = max{n EN:

a—an*nx? > 0} =1 and hence the set X.,1 defined in Lemma 3.2.5 is empty. Thus,

all assumptions in Theorem 3.2.7 are satisfied.

The feedback L?-gain functions k,(-) are given in Figures 4.3.2 and 4.3.4. From

these plots, it is easy to see that control action is concentrated on the location

of sensors. This phenomenon is natural, since the optimal control is obtained

to minimize the cost functional J defined by equation (3.2.9) whose first term

a) |I2 = []C2(t)|2s = 2, |2(2)[?, where 2; = 0.3, 0.5 and 0.75 for i = 1,2

and 3, respectively.

The corresponding closed-loop trajectories are shown in Figure 4.3.3 (for a = 0.3)

and Figure 4.3.5 (for a = 0.6). From Figures 4.3.3 and 4.3.5, we can see how the

controllers contribute to stabilization of the steep gradient as well as the solution

itself. Scheme 1 described in Section 4.2 was used for both cases.

EXAMPLE 4.3.2. (Identity Output Operator)

For this example, we take the identity operator J on L?(0,1) for output operator

C. In this case, the output space Y is L?(0,1). The convergence of gain functions and

65

corresponding closed-loop trajectories for a = 0.3 and 0.6 are shown in Figures 4.3.6-

4.3.7 and 4.3.8-4.3.9, respectively. Since the observation operator is the identity, this

example gives the information about maximal control action. We note the following

observation concerning the convergence rate of the gain function. Theoretically, the

rate is O(;), ([23, p15]). But, in this example, the rate seems to be faster than

O(#) (see Figures 4.3.2 and 4.3.4). Another observation is about the location of

maximal control action. The location moves to the left portion of domain as the

degree of stability a > 0 increases. In other words, we should put more action on

the front part of domain to get a higher exponential decay rate a > 0. (See also

Figures 4.3.2 and 4.3.4).

EXAMPLE 4.3.3. (Unbounded Observation)

Although we haven’t provided proof for the convergence of our approximation

schemes for the control problem with unbounded output operators, we give the

numerical evidence for the convergence of the feedback gain functions through ex-

amples. The convergence of the approximation scheme for the control problem with

unbounded input/output operators is an open problem.

Figures 4.3.10-4.3.17 are concerned with the control problems with unbounded

observation operator C € L(H4(0,1),R*°). Figures 4.3.10-4.3.12 show how control

action is related to the location of sensors. For these three cases, the number 0.3

is chosen for the desired degree of stability a > 0. Thus, n, = 1, Xq,1 is an empty

66

set and Y,, = { }}, (see Lemma 3.2.5). The output operators C’ for Figures 4.3.10,

4.3.11 and 4.3.12 are given by

C(z) = (2(0.3), (0.5), (0.75), 2/(0.2), 2’(0.4), 2’(0.6)), (4.3.5)

C(z) = (2(0.3), 2(0.5), 2(0.75), 2'(0.4), 2’(0.6), 2'(0.88)) (4.3.6)

and

C(z) = (2(0.3), 2(0.5), 2(0.75), 2/(0.3), 3(0.5), 3'(0.75)), (4.3.7)

respectively, where the 2(Z;)’s are defined by equation (4.3.4) and 2’(y;), 7 = 1,2, 3,

is defined by

to)=n [£40 0<$§<<1 (4.3.8) ZY; = 35 yb qe? S, . J.

From these examples, it is easy to observe that the control action depends

strongly on the location of sensors measuring the gradient of the solution instead of

the solution itself. Similar arguments in Example 4.3.1 can be used to explain this

phenomena. But, at this time, the first term in the performance index J consists of

two components:

3 3

Cz(e) I = D0 DP + D1), (4.3.9) t=1 t=1

where Y = R®. From equation (3.2.22) in Section 3.2, it is easy to see that the

second term )~*_, |2(g;)|? dominates the first term.

To see the control effects on the closed-loop solution (Figure 4.3.13), the gain

function in Figure 4.3.12 was chosen. For this case, we used Scheme 1 described in

Section 3.2.

67

Figures 4.3.14, 4.3.15 show the average of the gain function in Figure 4.3.12 and

the corresponding closed-loop solution (Scheme 2). The average values were esti-

mated on each interval (¥;_1,9;), where y; = 0, 0.3, 0.5, 0.75, 1.0 for 7 = 0,1, 2,3, 4,

respectively. Scheme 3 was applied to obtain Figure 4.3.16. From Figures 4.3.13,

4.3.15 and 4.3.16, one can observe that Schemes 2 and 3 work very well compared

with Scheme | for relaxation of the steep gradient of the solution. From a practical

point of view, Scheme 3 is the most “efficient” in the sense that we need only data

from sensors to calculate the control function u(t), yet produce similar effects (see

Section 4.2).

EXAMPLE 4.3.1. (Robustness)

In this example we show the robustness of the feedback controller showed in

Figure 4.3.2. The feedback controller is obtained from the control system (4.1.17)

with Re=60, a = 0.3, b(z) = e” and the ouput operator C defined by equation

(4.3.3).

Figures 4.3.1, 4.3.17, 4.3.19 and 4.3.21 show open-loop trajectories for Reynolds

numbers 60, 80, 100 and 120, respectively. The corresponding closed-loop trajecto-

ries are shown in Figures 4.3.3, 4.3.18, 4.3.20 and 4.3.22, respectively. The order N

of approximation is chosen as N = 32 for Re=60, 80, 100 and N = 64 for Re=120.

From these examples, it is easy to see that the feedback controller obtained for

Re=60 stabilizes the steep gradient of the solution for Burgers’ equation with var-

68

ious Reynolds numbers. However, we see that the sharp gradient is relaxed slowly

as Reynolds number increases (see Figures 4.3.3, 4.3.18, 4.3.20 and 4.3.22). Note

that, from Theorem 3.2.10, the closed-loop nonlinear system v(t) given by equation

(3.2.35) satisfies the stability property

ll oC) [lz << 2Mem™ |lvo(-)Ilx2 - (4.3.10)

Although the exponential decay rate a is independent of Reynolds number, the

constant M = M(a, Re) depends strongly on the Reynolds number, Re.

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4.4. Summary.

In this paper, we looked at a feedback control problem for a nonlinear equa-

tion, in particular, Burgers’ equation. The method consists of linearization of the

nonlinear equation. We used the linear quadratic regulator (LQR) problem to find

optimal feedback gains. The linearized equation is the heat equation. It was also

proved that, under appropriate selection of the input functions and “sensors”, the

LQR problem for the linearized problem is detectable and stabilizable. We then

analysed a “shifted quadratic cost” to construct gains which produce a fixed decay

rate. In particular, we showed that the closed-loop system satisfies the inequality

|| 2(#, 05 zo) [Ina S M(e)e™™ |I2ollx2,

where a > 0 does not depend on the Reynolds number, but M(e) does (Theorem

3.2.10).

We also developed a numerical scheme for computing the feedback functional

gains. For bounded inputs and bounded outputs, the convergence follows from well-

known results. Convergence for unbounded input and output is an open question.

Several numerical experiments were performed and the following observations

were made:

1) The functional gains depend strongly on the “sensor location” and their type.

92

For example, if the output operator C' is given by

C(z) = (2(.3), 2(.5), 2(.75)) ,

then the gain function is concentrated on the location of sensors, 0.3, 0.5 and 0.75

(Figures 4.3.2 and 4.3.4). On the other hand, for the unbounded output operator

C given by

C(z) = (2(.3), 2(.5), 2(.75), 2'(.4), 2/(.6), 2(.88))

the control action is concentrated on the points 0.4, 0.6 and 0.88 which are the

locations of sensors measuring the gradient (Figure 4.3.11).

2) Even for an unbounded output operator, the numerical convergence is ob-

served (Figures 4.3.10, 4.3.11 and 4.3.12). Thus it appears that the question of

convergence for such systems could be answered theoretically.

3) The closed-loop nonlinear system is stabilized (as predicted) by linear feed-

back laws. Moreover, the steep gradients (for « + 0) are smoothed out by feedback.

4) To test the “robustness” of the feedback control law, two experiments were

performed. First, we replaced the functional gain by its average over some intervals

and noticed that the performance was almost the same. Second, we obtained the

functional gain k(-) from the control system at the Reynolds number, Re=60, and

applied it to the closed-loop system

9 t,2)- 2 “tt tn) —-o(1 “fo t,s)d 4.4.1 Hilh®) = Roggatha)—alhedgeetha)—eF f Ms)elts)ds (4.41)

93

at Re=80, 100 and 120 (Figures 4.3.18, 4.3.20 and 4.3.22). Although the perfor-

mance was decreased, the system (4.4.1) was still stabilized and smoothed out.

These results provided some insight into the possibility of using linear feedback

laws for nonlinear distributed parameter systems. On the other hand, they raised

several interesting questions that need further study. We plan to:

(i) develop approximation schemes for linear regulator problems with unbounded

input and ouput operators;

(ii) investigate the “shock” stabilization problems for conservation laws (zero

viscosity) by boundary feedback controls.

REFERENCES

[1] B. D. O. Anderson and J. B. Moore, “Linear System Optimisation with Prescribed

Degree of Stability.” Proc. JEEE, Vol. 116, No. 12, (1969), 2083-2087.

(2] H. T. Banks and J. A. Burns, “Hereditary Control Problems: Numerical Methods

based on Averaging approximation”, SIAM J. Control and Optim., Vol. 16, No.

2, (1978), 169-208.

[3] H. T. Banks, J. A. Burns and E. M. Cliff, “Parameter Estimation and Identifi-

cation for Systems with Delays”, SIAM J. Control and Optim., Vol. 19, No. 6,

(1981), 791-828.

[4] H. T. Banks and K. Kunish, “The Linear Regulator Problem for Parabolic Sys-

tems.” SIAM J. Control and Optim., Vol. 22, No. 5, (1984), 684-698.

[5] J. M. Burgers, “Mathematical Examples Illustrating Relations Occuring in the

Theory of Turbulent Fluid Motion.” Trans. Roy. Neth. Acad. Sci., Amsterdam,

Vol. 17, (1939), 1-53.

[6] J. M. Burgers, “A Mathematical Model Illustrating the Theory of Turbulence”,

Adv. in Appl. Mech., Vol. 1, (1948), 171-199.

[7] J. M. Burgers, “Statistical Problems Connected with Asymptotic Solution of One-

dimensional Nonlinear Diffusion Equation”, in Statistical Models and Turbulence

(eds., M. Rosenblatt and C. van Atta), (1972), Springer, Berlin, 41-60.

[8] G. Chen, H. K. Wang and S. Weerakoon, “An Initial Value Control Problem for

Burgers’ Equation”, in Distributed Parameter Systems, (ed. F. Kappel, K. Kunish

and W. Schappacher ), Lecture Notes in Control and Information Sciences, Vol.

75, (1985), 52-76.

[9] W. C. Chin, “Pseudo-Transonic Equation with a Diffusion Term”, AJAA Journal,

Vol. 16, (1978), 87-88.

94

95

[10] J. D. Cole, “On a Quasi-Linear Parabolic Equation Occuring in Aerodynamics.”

Quart. Appl. Math., Vol. IX, No. 3, (1951), 225-236.

[11] J. B. Conway, A Course in Functional Analysis, Springer-Verlag, New York, 1985.

[12] R. F. Curtain, “Stability of Semilinear Evolution Equations in Hilbert Space”, J.

Math. Pures et Appl, Vol. 63, (1984), 121-128.

[13] R. Datko, “Uniform Asymptotic Stability of Evolutionary Process in a Banach

Space.” SIAM J. Math. Anal., Vol. 3, No. 3, (1972), 428-445.

[14] C. A. J. Fletcher, “Burgers’ Equation : A Model for All Reasons.” Numerical

Solution of Partial Differential Equations, J. Noye ed., North-Holland Publ. Co.,

(1982), 139-225.

[15] H. Fujita and A. Mizutani, “On the Finite Element Method for Parabolic Equa-

tions I: Approximation of Holomorphic Semi-Groups”, J. Math. Soc. Japan, Vol.

28, (1976), 749-771.

[16] J. S. Gibson, “The Riccati Integral Equations for Optimal Control Problems on

Hilbert Spaces”, SIAM J. Control and Optim., Vol. 17, (1979), 537-565.

[17] J. S. Gibson, “Linear-Quadratic Optimal Control of Hereditary Differential Sys-

tems: Infinite Dimensional Riccati Equations and Numerical Approximations”,

SIAM J. Control and Optim., Vol. 21, No. 1, (1983), 95-139.

[18] J. S. Gibson and I. G. Rosen, “Shifting the Closed-Loop Spectrum in the Optimal

Linear Quadratic Regulator Problem for Hereditary Systems”, Institute for Com-

puter Applications for Science and Engineering, NASA Langley Research Center,

Hampton, VA, ICASE Report 86-16, 1986.

[19] J. Glimm and P. Lax, Decay of Solutions of Systems of Nonlinear Hyperbolic

Conservation Laws, Amer. Math. Soc. Memoir, No. 101, A.M.S., Providence,

1970.

[20] W. D. Hayes, “Pseudo Transonic Similitude and First-Order Wave Structure”,

Journal of the Aeronautical Sciences, Vol. 21, (1954), 721-730.

96

[21] D. Henry, Geometric Theory of Semilinear Parabolic Equations, Springer-Verlag,

New York, 1981.

(22] E. Hopf, “The Partial Differential Equation u; + uu, = puss.” Comm. Pure and

Appl. Math, Vol. 3, (1950), 201-230.

(23] K. Ito, “Strong Convergence and Convergence Rates of Approximating Solutions

for Algebraic Riccati Equations in Hilbert Spaces”, LCDS/CCS Report #87-15,

Brown University, 1987.

[24] H. Kielhofer, “Stability and Semilinear Evolution Equations in Hilbert Space”,

Arch. Rat. Mech. Anal., Vol. 57, (1974), 150-165.

[25] I. Lasiecka, “Unified Theory for Abstract Parabolic Boundary Problems - A Semi-

group Approach”, Appl. Math. Optim, Vol. 6, (1980), 287-333.

[26] P. D. Lax, Hyperbolic Systems of Conservation Laws and the Mathematical Theory

of Schock Waves, CBMS-NSF Regional Conference Series in Applied Mathematics,

Vol. 11, SIAM, 1973.

[27] M. J. Lighthill, “Viscosity Effects in Sound Waves of Finite Amplitude”, in Surveys

in Mechanics (ed. G. K. Bachelor and R. M. Davies ), Cambridge University Press,

Cambridge, (1956), 250-351.

(28] V. P. Maslov, “On a New Principle of Superposition for Optimization Problems”,

Uspekhi Mat. Nauk, Vol. 42, No. 3, (1987), 39-48.

[29] V. P. Maslov, “A New Approach to Generalized Solutions of Nonlinear Systems”,

Soviet Math. Dokl, Vol. 35, No. 1, (1987), 29-33.

[30] T. Musha and H. Higuchi, “Traffic Current Fluctuation and the Burgers’ Equa-

tion.” Japanese J. of Applied Physics, Vol. 17, No. 5, (1978), 811-816.

(31] O. A. Oleinik, “Discontinuous Solutions of Nonlinear Differential Equations.”

Uspsekhi Math. Nauk, Vol. 12, (1957), 3-73, (English Translation Amer. Math.

Soc. Trans. , Series 2, 26, 95-172).

97

[32] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential

Equations, Springer-Verlag, New York, 1983.

[33] A. J. Pritchard and D. Salamom, “The Linear Quadratic Control Problem for Infi-

nite Dimensional Systems with Unbounded Input and Output Operators.” STAM

J. Control and Optimization, Vol. 25, No. 1, (1987), 121-144.

[34] S. M. Rankin, “Semilinear Evolution Equations in Banach Spaces with Application

to Parabolic Partial Differential Equations”, Preprint, 1989.

[35] D. L. Russell, Mathematics of Finite-Dimensional Control Systems: Theory and

Design, Marcel Dekker Inc., 1979.

[36] M. H. Schultz, Spline Analysis, Prentice-Hall, N.J., 1973.

[37] J. Smoller, Shock Waves and Reaction-Diffusion Equations, Springer-Verlag, 1983.

[38] J. Stoer and R. Bulirsch, Introduction to Numerical Analysis, Springer-Verlag,

New York, 1980

[39] J. A. Walker, Dynamical Systems and Evolution Equations, Theory and Applica-

tions, Plenum Press, New York, 1980.

[40] S. Weerakoon, An Initial Value Control Problem for the Burgers’ Equation, Ph.D.

Thesis, Department of Mathematics, Pennsylvania State University, December,

1984.

[41] G. B. Whitham, Linear and Nonlinear Waves, Wiley, New York, 1974.

[42] J. Wloka, Partial Differential Equations, Cambridge University Press, Cambridge, 1987.

[43] J. Zabczyk, “Remark on the Algebraic Riccati Equation in Hilbert Spaces”, Ap-

plied Math. Optim., Vol. 2, (1976), 251-258.

VITA

Sungkwon Kang was born in Taejeon, Korea on December 25, 1956. He gradu-

ated from Taejeon High School in 1975. He received the B.S. degree in Mathematics

Education and the M.S. degree in Mathematics from Seoul National University in

Seoul, Korea in 1979 and 1981, respectively. During 1981-1985, he worked at the

Mathematics Department of the Korea Air Force Academy, where he served as an

Assistant Professor while having the rank of Air Force Captain. He received his

Ph.D. in Mathematics from Virginia Polytechnic Institute and State University in

1990.

He is a member of the American Mathematical Society and of the Society for

Aéf »ougleor— Industrial and Applied Mathematics.

98