Feedback spreading control laws for semilinear distributed parameter systems

Feedback spreading control laws for semilinear distributed parameter systems

Khalid Kassara

Department of Mathematics, University Hassan II Ain Chok, P.O. Box 5366, Casablanca, Morocco Received 6 May 1998; received in revised form 15 March 2000

Abstract

The paper shows, for the rst time that a feedback spreading control law for semilinear distributed parameter systems with compact linear operator can be provided by a selection procedure which involves both the dynamics of the system and the property to be spread. In the case of ane dependence upon the control a minimum energy feedback spreading control law will be derived by using a parametrized constrained optimization technique along with some facts of set-valued analysis.

c

2000 Elsevier Science B.V. All rights reserved.

Keywords:

Distributed systems; Spreadability; Monotone solutions; Optimal control; Constrained optimization

1. Introduction and statement of the problem According to environmental motivations, the prob- lem of existence and determination of spreading con- trols in distributed parameter systems has already been stated in [10]. There, especially in the linear case a direct approach was used where the problem is re- formulated as a quadratic equation in Hilbert space, but unfortunately this did not lead to an easy imple- mentable algorithm. More recently in [11], one has approximated the problem as a rather unusual optimal control problem whose treatment leads to a sequence of sub-optimal feedback spreading controls which in- volve a parametrized Riccati dierential equation in innite dimension.

Nevertheless, the above approaches, though sophis- ticated and elegant, are only applicable to the lin- ear case and concern a rather restricted class of the properties to be spread. Of course, this precludes any

E-mail address:kassara@facsc-achok.ac.ma (K. Kassara).

application of the results to real processes which gen- erally are nonlinear. In most of the paper we study spreading controls for distributed parameter systems governed by the following semilinear abstract dier- ential equation:

˙

z + Az = f(z) + B(z)v; t¿0;

z(0) = z

; (1)

where −A is an unbounded linear operator with do- main dom(A) densely included in a separable Hilbert space Z ⊂ L

() for p¿1 and an open domain in the Euclidean space R

. The operators f and B(·), respectively, have images in Z and L(V; Z ) and are possibly nonlinear. V denotes another Hilbert space and stands for the space of controls. Throughout this paper we restrict ourselves to assume the following hypothesis:

H

( −A is the innitesimal generator of a compact C

semigroup (S (t))

on Z:

0167-6911/00/$ - see front matter c2000 Elsevier Science B.V. All rights reserved.

PII: S0167-6911(00)00036-0

Now, let ! be a set-valued map from dom(!) ⊂ Z with measurable images included in , then a spreading control may be introduced as follows:

Deÿnition 1.1. A measurable function v : [0; t

(→ V is called a !-spreading control for system (1) if there exists a solution z : [0; t

(→ dom(!) which satises the condition

The family (!( z(t))

is nondecreasing: (2) One of the most interesting and innovative topics in this area is the design of feedback spreading control laws having the following form:

v = (z): (3)

The objective of this paper is rst to present a unied approach to the analysis. The key idea is that due to condition (2) which characterizes a spreading control, the problem may be restated as one of existence of monotone solutions with respect to a preorder (see [1]

and the references therein), for the notion of mono- tonicity. One consequence is that the feedback spread- ing controls (3) can be provided once a selection procedure of the feedback map is done. The latter, de- noted by C

, consists of a set-valued map which is dened through a certain tangential condition which involves both the system and the property to be spread.

The above formulation is central in this paper be- cause it enables us to characterize all of the feedback spreading control laws. In addition, it will have ap- plications to several optimal control problems which involve the notion of spreadability such as those for- mulated in [11]. Namely, we can cite the problem of the design of slow spreads which may be solved by determining a minimum energy spreading control law or else by the problem of searching spreading controls which generate a maximum speed spread. The rst problem will be investigated in this paper by using a Lagrangian saddle point method.

The contents are structured as follows: Section 2 presents the mathematical tools to be used in the sub- sequent sections. Section 3 focuses on the problem of existence of feedback spreading control laws. Section 4 is concerned with the minimum energy feedback spreading control problem.

2. Basic results and deÿnitions

We begin by stating an adaptation of a recent re- sult by Chis-Ster [7] on the problem of existence of

monotone solutions with respect to a preorder. Let K be a subset of Z and P a set-valued map P:K 7→ K.

The symbol 7→ will be used for denoting set-valued maps in the remainder of the paper. Let the map P be a preorder on K, i.e., which satises z ∈ P(z) and y ∈ P(z) ⇒ P(y) ⊂ P(z) for each z ∈ K. Now let

z : [0; t

(→ K then it is said to be monotone with re- spect to the preorder P if

z(s) ∈ P( z(t)) (06t ¡ s ¡ t

): (4) Several authors have investigated the problem of exis- tence of monotone solutions of dierential inclusions with respect to a preorder (see [1,14,7]). Especially, the latter, based on some new viability results for the innite-dimensional case established in [6], has con- centrated on the case of compact semigroup under in- terest in this paper. Its main result when applied to the particular case of semilinear systems, easily leads to the statements below. Let D be a nonempty subset of Z, for a couple (y; z) ∈ Z × D consider the tangential condition

∀ ¿ 0; ∃0 ¡ h ¡ and p ∈ B(0; ) such that

S(h)z + h(y + p) ∈ D:

(5)

Here B(0; ) denotes the ball in Z whose radius is and centered at the origin. Then dene for each z ∈ D the following tangential subset as in [14] by

T

(z) = : {y ∈ Z | which satisfy (5)}: (6) Then we have the following result.

Theorem 2.1. Consider system (7)

˙

z + Az = g(z) (t¿0);

z(0) = z

; (7)

where g: dom(g) → Z may be a nonlinear operator.

Then besides hypothesis H

assume the following:

(i) The operator g|

is strongly–weakly continuous;

i.e.; maps strongly convergent sequences of K into weakly convergent sequences in Z .

(ii) The preorder P has a closed graph; i.e., the set graph(P) = {(z; y) ∈ K

|y ∈ P(z)}

is closed.

Then system (7) has monotone solutions with re- spect to the preorder P for all initial data z

∈ K if and only if

g(z) ∈ T

(z) for each z ∈ K: (8)

Remark 2.2. It should be useful to notice the follow- (a) The compactness hypothesis H ing:

is essential in Theorem 2.1 and involves a wide class of parabolic systems (see [3,5]).

(b) In the case of noncompact semigroup then we have to quote a weak tangential condition which is es- tablished by Chis-Ster [7]. This condition is hard to verify but it may be very useful for hyperbolic equations.

(c) Since the semigroup S (·) is of class C

it can be easily seen that if z ∈ K ∩ dom(A) the tangential condition (5) may be replaced by the following one:

∀ ¿ 0; ∃0 ¡ h ¡ and p ∈ B(0; ) such that

z + h(y − Az + p) ∈ D:

(9)

(d) We also have the formula T

(z)=

y ∈ Z | lim inf

h↓0

× d(S(h)z + hy; D)

h = 0

(10) for each z ∈ D, where d( y; D) = inf :

k y − xk for each y ∈ Z.

Now let us show the result below.

Lemma 2.3. Assume that D is a closed and convex subset of Z . Then the map

z ∈ D 7→ T

(z) ⊂ Z has closed convex values.

Proof. From Eq. (5) we easily get T

(z) = \

¿0

cl [

h∈(0;)

1

h [D − S(h)z];

then T

(z) is closed for each z ∈ D. To show its con- vexity let y; y ∈ T

(z) and ; ¿0 such that + =1.

It follows that

d(S (h)z + h(y + y); D)

= d((S (h)z + hy) + (S(h)z + h y); D):

Since the function y ∈ Z → d(y; D) is convex, we have

d(S (h)z + h(y + y); D)

6d(S(h)z + hy; D) + d(S(h)z + h y; D):

Now (10) implies the desired result.

Next we recall the concept of lower semicontinuity of set-valued maps (see [8]). The set-valued map Q : K 7→ Y; where Y is a metric space is said to be lower semicontinuous if for each z

∈ K and any sequence of elements z

converging to z

; for each y

∈ Q(z

), there exists a sequence of elements y

∈ Q(z

) that converges to y

.

The mapping : K → Y; where Y is another set, is said to be a selection of the map Q : K 7→ Y if (z) ∈ Q(z) for all z ∈ K.

It is of interest to cite Michael’s selection theo- rem [2], which states that any lower semicontinu- ous set-valued map with closed convex values has a continuous selection.

3. On the existence of feedback spreading controls Consider again the control system (1) with the state space Z and let !: dom(!) 7→ be a set-valued map which characterizes a property to be spread. Very of- ten, as should arise in real biophysical processes the initial data which generate a spread are unknown or partially given. This allows us to assign them to be belonging to a given subset S ⊂ dom !. Then a feed- back spreading control law may be dened as follows.

Deÿnition 3.1. A function : S → V is said to be a feedback spreading control law for system (1) if for all initial data z

∈ S the following statements hold:

(i) System (1) with v = (z) has a solution z with values in S.

(ii) The control v = ( z) denes a spreading control.

In the sequel of the paper we assume the following hypothesis:

H

(

= : {(y; z) ∈ S

| !(y) ⊃ !(z)}

is a closed subset in Z

which obviously requires that S is closed. Then we need to build the map T

as follows. For each couple (y; z) ∈ Z × S consider the tangential condition

∀ ¿ 0; ∃0 ¡ h ¡ and p ∈ B(0; ) such that

S(h)z + h(y + p) ∈ S (11) and

!(S (h)z + h(y + p)) ⊃ !(z):

Therefore, we can dene for each z ∈ S

T

(z) = : {y ∈ Z | (y; z) satises cond: (11)} (12)

as well as the feedback map below

C

(z) = : {v ∈ V | f(z) + B(z)v ∈ T

(z)}: (13) Hence, we are ready to prove the following result.

Theorem 3.1. Let : S → V be such that the map- ping f + B(·) is strongly–weakly continuous. Then is a feedback spreading control law if and only if it is a selection of the map C

.

Proof. First note that condition (2) means that

!( z(t)) ⊂ !( z(s)) (06t ¡ s ¡ t

; z

∈ S) and therefore it may be written as

z(s) ∈ P

( z(t)) (06t ¡ s ¡ t

; z

∈ S) provided that P

denotes the set-valued map P

(z) = : {y ∈ S | !(y) ⊃ !(z)}

for each z ∈ S; (14)

which of course denes a preorder on S. Conse- quently, according to Denition 1.1 a !-spreading control for system (1) may be characterized by the ex- istence of a monotone solution with respect to the pre- order P

. We can therefore apply Theorem 2.1 with K = : S; P =P

and g =f +B(·). In fact, the strong–

weak continuity assumption on f + B(·) and the fact that

coincides with the graph of the preorder P

implies respectively the conditions (i) and (ii) of Theorem 2.1. By Eqs. (6) and (12) we have

T

(z) = T

(z) for each z ∈ S (15) and thus the proof is completed.

Remark 3.2. Note that in the above theorem, there are no smoothness assumptions on f; B nor . Only f + B(·) must be strongly–weakly continuous.

Remark 3.3. Thus, a feedback spreading control may be obtained by choosing (z) in (3) as a selection from C

(z).

At this point we need to suppose that the map ! satises the condition

H

( !(y + y) ⊃ !(y) ∩ !( y); ∀y; y ∈ S and ; ¿0 : + = 1:

Then we can show the following existence result.

Proposition 3.4. Suppose the following conditions:

(i) The map T

is lower semicontinuous on S.

(ii) ∀(z; y) ∈ graph(T

); ∃v ∈ V such that B(z)v+

f(z) = y.

Then there exists a feedback spreading control law for system (1).

Proof. By conditions H

and H

the preorder P

of Eq. (14) has closed convex values then Lemma 2.3 and Eq. (15) imply that the map T

also has closed convex values. Since it is lower semicontinuous due to condition (i) by Michael selection theorem, see the end of Section 2, it possesses a continuous selection, say

: S → Z . Now by condition (ii) one is able to construct a mapping v

=

(z) such that B(z)v

+ f(z) =

for all z ∈ S. Consequently,

denes a selection of the map C

for which B(·)

+ f(·) is strongly–weakly continuous (because continuous) as required by Theorem 3.1. Therefore

is a feedback spreading control law for system (1).

4. Minimum energy feedback spreading controls This section is concerned with solving the control problem with a view to design slow spreads by seek- ing to minimize the action of the spreading control.

Based on the previous section where feedback spread- ing control laws are provided by a selection procedure of the map C

given in (13), the purpose is to solve the following parametrized optimization problem:

v∈C

min

kvk

(z ∈ S) (16)

whose solution

=

(z), if it exists, clearly will dene a minimum energy feedback spreading control law. Note that in the literature this selection procedure is referred to as the minimal selection procedure [1,2].

For convenience we denote by

the projector of the best approximation for a nonempty closed convex subset K of Z .

4.1. A preliminary result

In order to derive our main result, it is useful to begin by showing the following lemma.

Lemma 4.1. Let f ∈ Z and let T be a closed convex subset of Z . Let B ∈ L(V; Z) be a linear operator satisfying the following condition:

kB

k

¿mkk

( ∈ Z ); (17)

where m ¿ 0 and B

denotes the adjoint operator of B. Then the minimization problem

Bv+f∈T

min kvk

(18)

has a unique solution v

given by

v

= B

R

(y

− f); (19) where R : = BB

and y

is given by the ÿxed point equation

y

=

[(1 − R

)y

+ R

f] (20) for arbitrary small ¿ 0. Furthermore; y

is inde- pendent of .

Proof. Since C={v ∈ V | Bv+f ∈ T} is a nonempty closed convex subset in V problem (18) has a unique solution which is v

=

(0). To compute v

a proper method may be provided by the Lagrangian functional (see [12,13]):

L(v; y; ) =

kvk

+ hBv + f − y; i (v ∈ V; y ∈ T; ∈ Z ):

In fact, it can be easily shown that if (u

; y

;

) is a saddle point for L, i.e.

max

L(u

; y

; ) = L(u

; y

;

) = min

v∈V;y∈T

L(v; y;

) then u

is a solution of problem (18) and by unicity u

= v

. Now, since both L and T are convex the saddle point (v

; y

;

) is characterized by

@L

@v (v

; y

;

) = 0:

@L

@y (v

; y

;

); y − y

¿0 (y ∈ T);

@L

@ (v

; y

;

) = 0:

So that we get v

+ B

= 0;

h

; y − y

i60 (y ∈ T);

Bv

+ f = y

∈ T

and therefore in an equivalent way, we have v

=

−B

where

solves the following system:

−R

+ f = y

;

h

; y − y

i60 (y ∈ T; ¿ 0);

y

∈ T:

(21)

Now by (17) the operator R is invertible since it is symmetric and coercive then using

yields

v

= B

R

(y

− f);

y

=

[(1 − R

)y

+ R

f] ( ¿ 0):

In order to complete the proof, it remains to show that the mapping

: T → T

y →

[(1 − R

)y + R

f]

has a xed point for some ¿ 0. Indeed, we get k

(y) −

( y)k

6k(1 − R

)ek

=kek

− 2hR

e; ei +

kR

ek

; where

y; y ∈ T and e = y − y:

Now since the operator R

is coercive we have for some m

¿ 0,

hR

y; yi¿m

kyk

(y ∈ Z);

then it follows that k

(y) −

( y)k

6(1 − 2m

+

kR

k

)ky − yk

:

Therefore is a contraction for ¡ 2m

=kR

k

and thereby it has a unique xed point y

which belongs to S. In addition, by (21) it follows that y

is inde- pendent of .

4.2. The main result

Now we are in a position to prove the main result.

Theorem 4.2. Assume that the conditions H

–H

are to be fulÿlled. In addition suppose that

(i) For each sequence (z

)

⊂ S; (v

)

⊂ V and (

)

⊂ Z,

z

→

z and v

→

v ⇒ B(z

)v

→

B(z)v;

z

→

z and

→

⇒ B

(z

)

→

B

(z):

(ii) For each z ∈ S the operator B

(z) satisÿes the

coercivity condition as in Eq. (17) as follows

kB

(z)k

¿m

kk

( ∈ Z ); (22)

where the coecient m

¿ 0 is such that for each

¿ 0 there exists M ¿ 0 such that m

¿ M for

each z ∈ S; kzk ¡ .

(iii) The mapping f : S → Z is continuous.

(iv) The map T

is lower semicontinuous and has a strongly–weakly closed graph.

Let R(·)=BB

(·). Then there exists a unique feed- back spreading control law

given by

(z) = B

(z)R

(z)(

(z) − f(z)) (z ∈ S);

(23) where

=

(z) satisÿes the ÿxed point equation

=

[(1 − R

(z))

+ R

(z)f(z)] (24) for some ¿ 0. Furthermore; the mapping

: S → V is strongly–weakly continuous.

Proof. We have to solve for each z ∈ S the optimiza- tion problem

( min kvk

with constraints :

B(z)v + f(z) ∈ T

(z): (25) Since the map P

of Eq. (14) has closed convex values (due to H

and H

) by Lemma 2.3 the map T

(·) = T

(·) has closed and convex values.

Therefore, due to condition (22) the assumptions of Lemma 4.1 are satised with B : = B(z); f : = f(z) and T = : T

(z). Hence, problem (25) has a unique solu- tion v

=

(z) for each z ∈ S and thus expressions (23) and (24), respectively, come from (19) and (20).

Therefore,

stands for the minimal selection of the map C

(with minimum norm). To conclude that

is a feedback spreading control law it remains to show, in accordance with Theorem 3.1, that the map- ping f +B(·)

=

is strongly–weakly continuous.

Indeed, let (z

)

be a sequence with (strong) limit z ∈ S. Then for each z

, applying the optimality sys- tem (21) yields

R(z

)

(z

) = f(z

) −

(z

);

h

(z

); y −

(z

)i60 (y ∈ T

(z

));

(z

) ∈ T

(z

):

(26)

Now let y ∈ T

(z). We know that the map T

(·) is lower semicontinuous on S (see condition (iv)), therefore, there exists a sequence (y

)

which con- verges to y and satises

y

∈ T

(z

) (for each n):

Hence, by letting y : = y

in Eq. (26) we get h

(z

); R(z

)

(z

)i6h

(z

); f(z

) − y

i

(for each n)

and then by using (22) it follows that m

k

(z

)k

6 kB

(z

)

(z

)k

6 h

(z

); f(z

) − y

i (27) for each n. Consequently, by condition (ii) and the fact that the sequence (f(z

))

is bounded (due to (iii)) it comes that the sequence (

(z

))

is bounded. There- fore, it has a subsequence (

(z

))

which is weakly convergent to

∈ Z . Now, since h

(z

); f(z

) − y

i → h

; f(z)−yi (because f(z

)−y

→ f(z)−y strongly and

(z

) →

weakly) then we get by passing to the lim inf in expression (27)

lim inf kB

(z

)

(z

)k

6h

; f(z) − yi:

Therefore, due to the second part of condition (i) it follows that

kB

(z)

k

6 lim inf kB

(z

)

(z

)k

6 h

; f(z) − yi (28) for every y ∈ T

(z). Moreover, letting

= : f(z) − R(z)

using conditions (i) and (iii) yield

(z

) →

and by condition (iv) it comes that

∈ T

(z). Due to Eq. (28) we can conclude that the couple (

;

) satises the optimality system

R(z)

= f(z) −

;

h

; y −

i60 (y ∈ T

(z));

∈ T

(z)

and by unicity we get

=

(z) and

=

(z):

Therefore, the sequences (

(z

))

and (

(z

))

, re- spectively, are weakly convergent to

(z) and

(z).

Thus as desired, we proved that the mapping

is strongly–weakly continuous on S. To show that the mapping

is such it suces to note that

(z

) = −B

(z

)

(z

) for each n

and apply the second part of condition (i) with

=

−

(z

).

Remark 4.3. The proof of the above theorem is con-

structive in the sense that as a consequence we can

exhibit a sequence of sub-optimal feedback spreading

control laws. In fact, based upon the xed-point equa- tion (24) we can consider the sequence of mappings (

(·))

given on S by

(z) ∈ T

(z) (z ∈ S);

(z)

=

[(1 − R

(z))

(z) + R

(z)f(z)];

(29)

then by arguing as in the proof of the theorem, the mappings (

)

are strongly–weakly continuous on S. By Theorem 3.1 the feedback laws

(z) = B

(z)R

(z)(

(z) − f(z)) (z ∈ S) (30) dene a sequence of feedback spreading control laws which are pointwise convergent to the optimal law

.

5. Conclusion

In the present paper, we have described in a unied manner, how the problem of spreading control can be solved for a given semilinear distributed parameter system. The main tool we used was the well-known notion of monotonicity with respect to a preorder which led us to characterize a spreading control as a selection of the feedback map. Moreover, we stress that since the above map, under reasonable hypothe- ses, has closed convex values then an important consequence is that several optimal spreading control problems may be investigated by using the classical optimization methods. As for instance in Section 4 a minimum energy spreading control law is derived.

Natural directions for further work include

• The study, by the same approach, of the maximum speed spreading control problem as stated in [11].

• The study of the case where the hypothesis H

does not hold as in semilinear hyperbolic systems. Its im- portance resides in the fact that numerous processes in which one can observe the spreading phenomena are of hyperbolic kind (see [4,9]).

• Application of the results established to real-word processes as pollution systems and vegetation dy- namics which gave rise to the notion of spreadabil- ity in [11].

Acknowledgements

The author wish to thank an anonymous referee for his helpful suggestions which improve the paper.

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