Control Reconstruction for Nonlinear Parabolic Equations

Working Paper

Control Reconstruction for Nonlinear Parabolic Equations

V. I. Maksimov

WP-94-04 October, 1994

'81 1 1 ASA

International Institute for Applied Systems Analysis A-2361 Laxenburg Austria r n ~ A

.I.I. Telephone: +43 2236 807 Fax: +43 2236 71313 o E-Mail: info~iiasa.ac.at

Control Reconstruction for Nonlinear Parabolic Equations

V. I. Maksimov

WP-94-04 October, 1994

Working Papers are interim reports on work of the International Institute for Applied Systems Analysis and have received only limited review. Views or opinions expressed herein do not necessarily represent those of the Institute or of its National Member Organizations.

FllASA

International Institute for Applied Systems Analysis A-2361 Laxenburg Austria

ih.43

Telephone: +43 2236 807 Fax: +43 2236 71313 E-Mail: info@iiasa.ac.at

Control Reconstruction for Nonlinear Parabolic Equations

Introduction.

The problem of reconstruction of a control for nonlinear parabolic equations is con- sidered. Regularized solution algorithms stable with respect to informational and com- putational disturbances are constructed. The suggested algorithms are guided from the theory of closed-loop control [1,2]. Inaccurate measurements of current phase states are the inputs, and approximations to the real controls are the outputs for the algorithms.

The algorithms are operating in real time. Estimations for convergence rate are provided.

The constructions are based on the approach of [8-121. T h e problem belongs t o the class of inverse problems of dynamics that are being inversely studied today (see, for example, investigations [3,6,7], where the corresponding bibliography is given). These problems considered as stationary (operator) ones were deeply investigated by many authors. In the present paper solutions of control reconstruction problems are built with the help of the method of feedback control with a model from the theory of differential games [1,2]

and the dynamical discrepancy method [4,5]. For other dynamical control reconstruc- tion algorithms based on the principle of feedback for distributed-parameter systems see [13-181.

1. Control with a Model.

Let us first consider the problem of reconstruction of a control for a parabolic equation in a Hilbert space (H,

I . I),

^{H =} H*,

where T = [to,#] is the time interval, and A : D(A)

c

H + 2H is the rn-accretive operator, i.e. the operator wich satisfies the following conditions:

a) for any xi E D( A ) and y, E Ax i, j = 1 , 2

b) R ( I

+

^{XA) = H VX}

>

0.

We also assume the following notations:

( a , .) ^- the scalar product in H;

D(A) - the domain of A;

D(A) ^- the closure of D(A);

f (.) E L2(T; H) - a given disturbance;

B E _L(U; H) ^- a linear continuous operator;

(U,

I . 1")

^- an uniformly convex real Banach space.

Definition 1.1. [19,20] A function

t + x(t) ⁼ x(t; to, xo, u(.)) : T + D(A), x(to) = xo, is called an integral solution of inclusion (1.1) on T if

a ) x(.) E C ( T ; H ) , x(t) E D(A) for each t E T , b) x(-) satisfies the inequality:

+

^{2 (x(T) - I , Bu(.r)}

+ ^f

^{(T) - y ) d ~ (1.2)}

s

Vx E D(A), y E Ax, t o < s < t < d .

For any u ( - ) E L2(T; U), f (-) E L2(T; H ) there exists a unique integral solution of (1.1) (Theorem of Benilan, see Lakshmikantham and Leela, [20, Theorem 3.5.1, p. 104

]

).

Along with the general case, where A is an arbitrary multivalued m-accretive operator and the set Ax is convex for any x E D(A), consider the special case with the operator A of the form:

Ax ⁼ A,x for x E D(A) = {x E V : A,x E H ) . (1.3) Here A, : V + V* is the single-valued operator such that:

1) A, is Lipschitz, semicontinuous and monotone; A,O = 0;

2) (A,x ^- A,y, x ^- y)

>

^{wllx - y1I2} for all x, y E V, where w

>

0 is a fixed constant;

3) \IA,xllv*

5

C ( l

+

11x11) for all x E V, where C does not depend on x;

hereafter (V,

(1 .

11) is a real separable and reflexive Banach space such that V

c

H and the inclusion mapping of V into H is continuous, and (-, -) is the duality between V and v * .

In this (special) case we assume xo ^E H . Then there exists a unique strong solution x(.) = x ( - ; to, xo, u(.)) of (1.1), defined as the function which satisfies (1.1) and such that x(.) E L2(T; V), it(.) E L2(T; V*) (see Lions [21, Theorem 1.2, p.1731 ).

Notice that the operator A of the form (1.3) is m-accretive (see Vrabie [22, Theorem 6.31 ).

An example of an operator A, is A, : V = H,'(R) + V* = H-'(0) of the form:

Here R

c

Rh is a bounded simply connected domain with the sufficient?, ^. brlooth bound- ary; H,'(R) and H-'(0) are standard Sobolev spaces 121, Lions]; ? ,! is the gradient of the convex differentiable function j : R +

R+

= {r E R : r _> 0), Al : V + V*,

If the function r + P ( r ) is Lipschitz and P(0) = 0, then conditions 1) - 3) hold.

Denote: XT ^- the bundle of solution on (1.1), i.e.

in the special case ( h ) ⁼

{ ^:'+

6 ( h ) ( l

+

O(Ah)), in the general case.

Here

The problem in question can be explained as follows. The integral solution x,(.) ⁼ x(., to, xo, u,(.)) of the system (1.1) (strong solution in special case) depends on a time- varying unknown control

uT(.) E UT ⁼ {u(.) E LZ(T; U) : u(t) E P for a.a. t E T),

where P

c

U is a convex, closed and bounded set. The interval T is put into parts by intervals [ T ; , T ; + ~ ) , i E [0 ^: m ^- 11, ^{~ i + l} = ~i

+

^{6, 6}

^>

^{0, TO = to, T~ = 29.} At time instants

T; E A ⁼

{7;)z0

^{the x,(T;)} are measured approximately, i.e. the elements

th,;

^{E H close}

x,(T;) are found:

l x T ( ~ i ) - (h,;

1 ⁵

^h-

Here h is the level of the informational noise. A solution x,(.) is unknown. Let U,(x,(-)) be the set of all controls with values in UT, generating the x,(-). The problem is t o calculate an approximation to an element u,(.) = u(.; _x,(.)) E U,(x,(.)) syncronically with the motion process, basing on inaccurate measurements of x,(T;).

To calculate approximately u,(-), we use the method of closed-loop control with a model [ I , 2, 8-18]. According to this approach, the problem of reconstruction of an unknown control through the measurement results

thy;

is substituted by the problem of positional control for an auxiliary system M ( a model). Therefore, the problem of reconstruction of u,(-) is subdivided into:

i ) the problem of selecting a model (functioning synchronically with the real system), ii) the problem of guiding a model through a positional control algorithm.

In [15] it was noticed that for some cases of parabolic systems a copy of the real system can serve as a model. Examples of such concrete systems were provided. We take M as a copy of the real system (1.1) for the system equations with the m-accretive operator.

This copy is:

w(t)

+

^{Aw(t) 3} ~ v ~ ( t )

+

^{f (t), t E T,} (1.4) Describe the above mentioned algorithm.

First, a family A h = { T ~ , ; ) ? ~ , ^{T ~ , O} = to, ~ h , ~ , , = 29, ~ h , i = ~ h . , i - ~ - 6(h) of partitions of the interval T with diameters 6(h), and a function o ( h ) are fixed. The functions 6(h) and o ( h ) are chosen so as:

Before the initial time of the process, values h, o ( h ) and a partition A = { A h ) z o , m = mh are fixed. The work of the algorithm staring at time to is decomposed into m-1

identical steps. At the i-th step run during the time interval S; = [T,, T ; + ~ ) , T; = rh,;, the following operations are carried out. At time T; a control

v (t) h = vf for a.a. t E

&,

vi h = arg min{l,(s;, v) : v E P), _(1.5) l o ( ~ i , ²⁾⁾ = 2(si,v)

+

~ ( h ) ) v l L , si = ~ ( 7 i ) - [ h , i

is calculated. A control vh(t) is a piece-wise constant function formed by the feedback:

Finally, the state w(T;) of the model (1.4) is transformed into W ( T , + ~ ) . The procedure stops at time 8.

Let u,(.) = u,(.; xT(.)), xT(.) E XT, be the element of the set U,(xT(-)) whose L2(T; U)- norm is minimal.

Theorem 1.1. Let - A generate the nonexpansive compact (Vt

>

0) semigroup S ( t ) . In the general case

Proof. From the inequality (1.2), Benilan's [20, Theorems 3.6.1 and 3.5.11 and Vrabie's [23,Theorem 21 result follow the propositions:

a ) For every t,,t* E T , t,

<

t*, x, E H, u(.) E Ut,,t* = {u(.) E Lz([t,,t*];U) : u ( t ) f P for a.a. t E [t,, t*]} the solution x(-; t,, x,, u(.)) is continuous.

b) (semigroup property) For every t* E T , t* E (t,, 81, uto,t. ^{( a )} E Uto,tr, ut.,t* (-) E Ut,,t*

and t E [t,, t*] the equality

is true, where x, = x(t,; to, xo, uto,t*(.)),

u ~ ~ , ~ . ( - ) for a.a. 7 E [to, t,]

ut0,t*(t) =

{

u ~ . , ~ * ( - ) for a.a. T E [t,, t*].

C) For every t,,t* f T , t,

<

t*, xl,x2 E H, ul(.),u2(.) E Ut,,t* functions T ^-+

L ( x ~ ( T ) , u ~ ( T ) ) : [t,, t*] ^-+ R, L(x, U ) = 2(x,u), where j = 1,2, x j ( - ) = x(., t , , x j , u j ( - ) ) are summable and

( ~ l ( t * ) - x2(t*)12

5

<xl(t*) - x2(t*))12+

t*

+ J

L ( ~ l ( 7 ) - x2(7), ~ l ( 7 ) ) - u2(7))d7,

t

d) The set U,(xT(.)) is convex, bounded and closed in L2 (T; U).

e) The bundle of motion X T is uniformly bounded and semicontinuous in C ( T ; H).

The proof of the theorem follows from the result of [15].

Consider the special case. Let the following condition hold.

Condition 1.1. Let U = V, V be a real Hilbert space, B be the canonical embedding of V into H, Xb C X T be the set of all solutions of the system (1.1) such that for any xT(.) f X b the full variation of the control u,(.) = u,(.; xT(.)) E UT generating x T ( - ) is bounded by a b E (0, +m).

Then the following theorem is true.

Theorem 1.2. In the special case, under Condition 1.1, the following estimation for the convergence rate is true:

where k = k(Xb) = const E (0, ^+m).

The constant k is found explicitly. The proof of Theorem 1.2 follows from Lemma 1.1.

Lemma _{1.1. Let} Y be a Banach space, vl E L,(T; Y*), v2(t) E Y Vt E T be a function with the bounded variation,

Then the bound

B

holds.

Here (-, .)y is the duality between Y and Y*, varyv2(.) stands for the full variation of a function v2(.) on the time interval T .

Proof. The proof of the lemma is analogous to that of Lemma 1 [24]. Let ~k + 0 as k + w ,

It holds

Here

7

! = Thk,i, mk = mh,. Due to the inequality I v ~ ( - ) I ~ , ( ~ ; ~ . )

<

po

<

m, we have

Note that

5 {

^k-tm

I C

( ~ - ( T : ) , V ~ ( T ~ ~ ) - v Z ( ~ : ) ) y

+

i=l

<

^{l i m y k-tm}

C IV~(T:-~)

^{- vt(r:)ly}

+

^yd

<

i=l

the lemma is proved.

Proof of Theorem 1.2. It is easy t o establish the relations:

k j E ( 0 , +oo), j E [ l : 71.

Here

@;(Ah; X T ( ' ) ) = s u ~ { ( I x r ( ~ h , i ) - X T ( T ) ~ ~ V * : E [ ~ h , i , ~ h , i + l ] ) ,

&h(t) = sup ~ X T ( T ) - w(T)l2+

~ € [ t o , t ] t

+ a ( h )

I{

^{lvh((r)l$ -} lu,(.; x T ( . ) ) l$}dr.

t o

Therefore

t

&h(t)

+

^u

J

^{ljxr(7) -} W ( T ) I J ~ ~ T

<

^ks(h

+

b ( h ) ) , t E T .

t o

Consequently, Vt 1 , t 2 E T , t

<

t 2

u*(.) ⁼ u*(.; x,(-)).

Note that Lemma 1.1 and inequality (1.6) imply the inequallity

5

k{h

+

^&(h)

+

^a(h)'J2

+

^(h

+

&(h))/a(h)}, where k = k(xb). The theorem is proved.

If there exists a test function t ^t uo(t) = uo E U, t E T, and one has t o compute an element from U,(X,(.)) nearest in L2(T; U) to uo, then in (1.5)

2. Dynamical Discrepancy Method.

In this section we consider a dynamical modification of the discrepancy method from the theory of ill-posed problems [4,5] and provide upper and lower bounds for its convergence rate. Denote: V and H ^- real Hilbert spaces, V* and H* ^- spaces dual t o V and H respectively, ^{( a ,} .) ^- the duality between V and V*, (., .) ^- the scalar product in H,

11 . )I

and

I - 1

^- the norms in V and H respectively, P

c

H ^- a convex, bounded and closed bundle. We identify spaces H and H* and suppose that V is densely and continuously imbedded in H. Let A h be a partition of the interval T with the diameter &(h).

Consider a parabolic system whose evolution is described by

i ( t )

+

^{Alx(t) = u(t)}

+

^{f ( t ) ,} for a.a. t E T, _(2.1) x(to) = xo.

Here Al = A

+

^{A2, A : V t} V* is a linear, continuous and symmetric operator satisfying the coercivity condition

for certain w

>

^{0 and a} E R; A2 is the gradient of a convex differentiable function p : H ^-t

R

⁼ [O, ^$001,

j(.)

E W1tW(T; H) = { o ( . ) E L2(T; H) : vt(.) E L,(T; H)).

The problem considered is analogous to that described in Section 1. An unknown control u, = us(.; x,(.)) E UT acts upon the system (2.1) and generates a motion x,(.).

At timt. instarii,~ 7; = rh,; t Ah, the history x~,-,,,~(-) of the motion is measured approxi- mately, i.e. a piece-constant function = ,, ^{( a )} being an approxirnation of xTi-, ,, ^{( a )} is memorized:

An algorithm for computing an approximation to us(-) is to be found.

Assume the following

Condition 2.1. xo E D(AH) = {x E V : Ax E H), u,(.) E UT ⁼ {v(.) E L2(T; H) : v(t) E P, J6(t)l

5

lil for a.a. t E T), I<

<

+oo, and the operator A2 is Lipschitz.

By Theorem 4.3 [25] (see also [26]), for every u(.) E UT there exists a unique solution x(.) = x(.; to, xo, u(.)) of (2.1) such that x ( - ) E W1*"(T; H)

n

C ( T ; V), t ^H Ax(t) E L 2 ( T ; H ) . Introduce the sets

h,6 h h

<

k(h6-' $ S)),

vk,,

^{(pi ) = { U E P :}

1~

^- $ ( ~ i - l , ~ i ;

tTi-l,Ti(.))I

^-

where

Let the partition A h be such that

Describe the desired algorithm. Before the initial time of the process, values h, k and the partition A ⁼ A h are fixed. The work of the algorithm starting at time to is decomposed into mh-1 steps. At the i-th step run out during a time interval S; = Sh,; = [ T ~ , ; , ~ h , i + l ) , a control vh(t) = vh(t; x,(.)) = v;h (t E Sh,;), i

2

1,

h 6 h

h argmin{lul: U E Vk; ( p , ) ) , if ~[f(p:)

# 0

V; =

0, in the opposite case, v,h = arg min{lul : u E P), is calculated. The procedure stops at time 29.

Denote : XT ^- the bundle of motion : XT ⁼ {x(.; to, x0, u(.)) : u(.) E UT); u,(-; x(.))

- a control generating a motion x(.) E XT; E(x,(.), h) ^- the set of all functions

th(.)

^{: T -+} D ( A h ) such that the inequalities (2.2) hold; (E(x,(.), h) ^- the set of all measurement results admissible for x,(.));

x(.) E XT,

th(-)

E Z ( x ( - ) , h ) ) .

Theorem 2.1. There exists a k,

2

0 such that for every k E [k,, +a)

The constant k, is found explicitly.

The proof of Theorem 2.1 is analogous to that of Theorem 1 [18].

So, if values h, S(h) and h6-'(h) are "sufficiently small", then vh(.) ⁼ vh(.;x,(.)) is a "good" approximation to ti,(-) = u,(-; x(.)).

Theorem 2.2. Let i n t P

# 0.

Then for h + 0 the following estimations for the convergence rate are true:

Note that some upper estimations for the convergence rate of dynamical algorithms of control reconstruction are obtained in [24] for systems described by ordinary differential equations, and in papers [14, 271 for systems described by partial differential equations.

Proof of Theorem 2.2. Let x ( - ) E XT, h ( - ) E (0, I ) ,

th(.)

^E Z ( x ( - ) , h). Using the local Lipschitz property of the mapping A2, one can easily deduce the inequality

+

^{A 2 4 ) -} f ( t ) ) d t (

I

^ki(h

+

^b2)-

Then for k

2

kl, i E [l : m - 11, m = mh

h 6 h

and, consequently,

Vk,;

(pi )

# ^0.

Note that (2.3), (2.5) imply the inequality

Further, from (2.6) we have

By (2.7) and (2.8) we deduce

This and Lemma 2.1 imply for k

>

kl, h E (0, I.)

The upper bound (2.4) is proved. Obtain the lower bound. Let vo E i n t P , lvol

#

0, ,B

>

0 be such that

Note that

u(.; x(.)) = vo(.)

if vo(t) = vo Vt E T, x(.) = x(.; xo, vo(.)). In order to prove the lower bound (2.4), it is sufficient to show that for small h, 6(h), h6-'(h) we have

uniformly with respect to all Ah, t h ( . ) E Z(x(.), h). Here the constant kg does not depend on h, 6(h). Further, we have

Consequently,

1'$(7i-lr R, $ - I , T i ( . ) ) -

~ o I 5

k ~ ( h / b

+

^6).

Let k

>

^{k, =} 2max{k1, k5). Then from (2.9) we have

h,6 h

if dl = min{p, kS(h6-l

+

6)). Since v: is the minimum-norm element from Vk,, (pi ), so v: $! S(vO; dl 12). Therefore

if 6 and h6-' are sufficiently small, i.e. h E (0, h,), where h, is such that k5(h6-l(h)+

6(h))

5

,O for h E (0, h,). From (2.10), (2.12) we deduce (2.1 1). The theorem is proved.

Let P = {u E U : lul

5

m ) , m E (0, +m). Then by virtue of the equality (2.3) the element v: is determined by the formula [28, Example 1.4.1.1 :

Pi-d.pi/Ip;I, if

IP;I

^{> d ,}

V; =

0, in the opposite case, where

d = k(h6-'

+

^6),

pi

⁼

(t" t;-'))s-' + ^'$;h +

^{A,(;-, -}

^f

^(7;-1).

3. Example.

In this example we illustrate the algorithm of Section 2. Let

For these data, an approximation v h ( t , 7 ) to u ( t , 7 ) was computed by a net method with the usage of formula (2.3), (2.4). The domain R was replaced by the net with step 1 / N , N = 15. The space H and the sets

v:f(p:(.))

were replaced by the net space HN and the sets ~ ; " : ' ~ ( p , h ( . ) ) , , respectively :

i-1

-6

C

a t h ( r j - 1 , qjr)

+

p ( t h ( T i , ~ j r ) ) -

f

('i, % r ) ,

j=1

7 . _3' = j r ~ - ~ , j,r E [ l : N ] . Here A is the Laplace operator.

Figs. 1-5 show the cross-sections of the surfaces u ( t , 7 ) and v h ( t , 7 ) for t h ( t ) =

x ( t )

+

^ht.

The present research was supported partly by International Science Foundation Grant No. NMDOOO and Grant No. 93-011-16129 of the Fund for Fundamental Research of the Russian Academy of Sciences.

C 0 0 0

er

fl II II

u

P P.

a x e

Fig. 3

DISCREPRNCY METHOD

t

XI

=

0 . 4 0 x2

=

^{0 .GO}

CONTROL :

- - R e a l

-

^{M o d e 1}

S t r u c t u r e o f n o i s e i n the m o d e l : eps*t

1

Fig. 5

D I S C R E P f i N C Y METHOD

xl

=

0 . 4 0 x 2

=

^{0 . 7 0}

- - R e a l

-

^{Mode 1}

h

S t r u c t u r e of noise in the model

:

eps*t 2

-

) t

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V. I. Maksimov.

Institute of Mathematics and Mechanics, Urals Branch, Academy Sci. of Russia, Kovalevskoi 16,

620219 Ekaterinburg, Russia.