Open access peer-reviewed chapter

On the Stabilization of Infinite Dimensional Semilinear Systems

By El Hassan Zerrik and Abderrahman Ait Aadi

Submitted: March 16th 2019Reviewed: May 26th 2019Published: November 27th 2019

DOI: 10.5772/intechopen.87067

Downloaded: 41

Abstract

This chapter considers the question of the output stabilization for a class of infinite dimensional semilinear system evolving on a spatial domain Ω by controls depending on the output operator. First we study the case of bilinear systems, so we give sufficient conditions for exponential, strong and weak stabilization of the output of such systems. Then, we extend the obtained results for bilinear systems to the semilinear ones. Under sufficient conditions, we obtain controls that exponentially, strongly, and weakly stabilize the output of such systems. The method is based essentially on the decay of the energy and the semigroup approach. Illustrations by examples and simulations are also given.

Keywords

  • semilinear systems
  • output stabilization
  • feedback controls
  • decay estimate
  • semigroups

1. Introduction

We consider the following semilinear system

żt=Azt+vtBzt,t0,z0=z0,E1

where A:DAHHgenerates a strongly continuous semigroup of contractions Stt0on a Hilbert space H, endowed with norm and inner product denoted, respectively, by .and .., v.Vad(the admissible controls set) is a scalar valued control and Bis a nonlinear operator from Hto Hwith B0=0so that the origin be an equilibrium state of system (1). The problem of feedback stabilization of distributed system (1) was studied in many works that lead to various results. In [1], it was shown that the control

vt=ztBzt,E2

weakly stabilizes system (1) provided that Bbe a weakly sequentially continuous operator such that, for all ψH, we have

BStψStψ=0,t0ψ=0,E3

and if (3) is replaced by the following assumption

0TBSsψSsψdsγψ2,ψHforsomeγT>0,E4

then control (2) strongly stabilizes system (1) [2].

In [3], the authors show that when the resolvent of Ais compact, Bself-adjoint and monotone, then strong stabilization of system (1) is proved using bounded controls.

Now, let the output state space Ybe a Hilbert space with inner product ..Yand the corresponding norm .Y, and let CLHYbe an output operator.

System (1) is augmented with the output

wtCzt.E5

The output stabilization means that wt0as t+using suitable controls. In the case when Y=Hand C=I, one obtains the classical stabilization of the state. If Ωbe the system evolution domain and ωΩ, when C=χω, the restriction operator to a subregion ωof Ω, one is concerned with the behaviour of the state only in a subregion of the system evolution domain. This is what we call regional stabilization.

The notion of regional stabilization has been largely developed since its closeness to real applications, and the existence of systems which are not stabilizable on the whole domain but stabilizable on some subregion ω. Moreover, stabilizing a system on a subregion is cheaper than stabilizing it on the whole domain [4, 5, 6, 7, 8]. In [9], the author establishes weak and strong stabilization of (5) for a class of semilinear systems using controls that do not take into account the output operator.

In this paper, we study the output stabilization of semilinear systems by controls that depend on the output operator. Firstly we consider the case of bilinear systems, then we give sufficient conditions to obtain exponential, strong and weak stabilization of the output. Secondly, we consider the case of semilinear systems, and then under sufficient conditions, we obtain controls that exponentially, strongly, and weakly stabilize the output of such systems. The method is based essentially on the decay of the energy and the semigroup approach. Illustrations by examples and simulations are also given.

This paper is organized as follows: In Section 2, we discuss sufficient conditions to achieve exponential, strong and weak stabilization of the output (5) for bilinear systems. In Section 3, we study the output stabilization for a class of semilinear systems. Section 4 is devoted to simulations.

2. Stabilization for bilinear systems

In this section, we develop sufficient conditions that allow exponential, strong and weak stabilization of the output of bilinear systems. Consider system (1) with B:HHis a bounded linear operator and augmented with the output (5).

Definition 1.1 The output (5) is said to be:

  1. 1. weakly stabilizable, if there exists a control v.Vadsuch that for any initial condition z0H, the corresponding solution ztof system (1) is global and satisfies

CztψY0,ψY,ast,

  1. 2. strongly stabilizable, if there exists a control v.Vadsuch that for any initial condition z0H, the corresponding solution ztof system (1) is global and verifies

CztY0,ast,

and

  1. 3. exponentially stabilizable, if there exists a control v.Vadsuch that for any initial condition z0H, the corresponding solution ztof system (1) is global and there exist α,β>0such that

CztYαeβtz0,t>0.

Remark 1. It is clear that exponential stability of (5)strong stability of (5)weak stability of (5).

2.1 Exponential stabilization

The following result provides sufficient conditions for exponential stabilization of the output (5).

Theorem 1.2 Let Agenerate a semigroup Stt0of contractions on Hand if the condition:

  1. ReCCAyy0,yDA, where Cis the adjoint operator of C,

  2. CStyYαCyYand CByYβCyY, for some α,β>0,

  3. there exist T,γ>0such that

0TCCBStyStydtγCyY2,yH,E6

hold, then there exists ρ>0for which the control

vt=ρsignCCBztzt

exponentially stabilizes the output (5).

Proof: System (1) has a unique mild solution zt[10] defined on a maximal interval 0tmaxby the variation of constants formula

zt=Stz0+0tvsStsBzsds.E7

From hypothesis 1, we deduce

ddtCztY22ρCCBztzt.

Integrating this inequality, we get

CztY2Cz0Y22ρ0tCCBzτzτ.E8

It follows that

CztYCz0Y.E9

For all z0Hand t0, we have

CCBStz0Stz0=CCBztztCCBztztStz0+CCBStz0ztStz0.

Using hypothesis 2 and (9), we have

CCBStz0Stz0CCBztzt+2ραβCztStz0YCz0Y.

It follows that from (7) and condition 2 that

CCBStz0Stz0CCBztzt+2ρα2β2TCz0Y2.E10

Integrating (10) over the interval 0Tand replacing z0by ztand using (6), we deduce that

γ2ρα2β2T2CztY2tt+TCCBzszsds.E11

It follows from the inequality (8) that the sequence CznYdecreases and that for all nN, we have

CznTY2Czn+1TY22ρnTn+1TCCBzszsds.

Using (11), we deduce

CznTY2Czn+1TY22ργ2ρα2β2T2CznTY2.

Taking 0<ρ<γ2α2β2T2, we get

CznTY22ρ1+2ργ2ρα2β2T2Czn+1TY2.

Then

CznTY21MnCz0Y2.

where M=1+2ργ2ρα2β2T2>1.

Since CztYdecreases, we deduce that

CztYMelnM2Ttz0,t0,

which gives the exponential stability of the output (5).

Example 1 On Ω=]0,1[, we consider the following system

zxtt=Azxt+vtzxtΩ×]0,+[zx0=z0xΩ,E12

where H=L2Ωand Az=z. The operator Agenerates a semigroup of contractions on L2Ωgiven by Stz0=etz0. Let ωbe a subregion of Ω. System (12) is augmented with the output

wtχωzt,E13

where χω:L2ΩL2ω, the restriction operator to ωand χωis the adjoint operator of χω. Conditions 1 and 3 of Theorem 1.2 hold, indeed: we have

χωχωAyy=χωyL2ω20,yL2Ω,

and for T=2, we have

02χωχωBetyetydt=02e2tdtωy2dx=1212e4χωyL2ω2.

We conclude that for all 0<ρ<e4116e4, the control

vt=ρifχωztL2ω20,0ifχωztL2ω2=0,

exponentially stabilizes the output (13).

2.2 Strong stabilization

The following result will be used to prove strong stabilization of the output (5).

Theorem 1.3 Let Agenerate a semigroup Stt0of contractions on Hand B:HHis a bounded linear operator. If the conditions:

  1. ReCCAψψ0,ψDA,

  2. ReCCBψψψ0,ψH, hold, then control

vt=CCBztzt1+CCBztzt,E14

allows the estimate

0TCCBSsztSsztds2=Ott+TCCBzszs21+CCBzszsds,ast+.E15

Proof: From hypothesis 1 of Theorem 1.3, we have

12ddtCztY2RevtCCBztzt.

In order to make the energy nonincreasing, we consider the control

vt=CCBztzt1+CCBztzt,

so that the resulting closed-loop system is

żt=Azt+fzt,z0=z0,E16

where

fy=CCByy1+CCByyBy,forallyH

Since fis locally Lipschitz, then system (16) has a unique mild solution zt[10] defined on a maximal interval 0tmaxby

zt=Stz0+0tStsfzsds.E17

Because of the contractions of the semigroup, we have

ddtzt22CCBztztBztzt1+CCBztzt.

Integrating this inequality, we get

zt2z0220tCCBzszsBzszs1+CCBzszsds.

It follows that

ztz0.E18

From hypothesis 1 of Theorem 1.3, we have

ddtCztY22CCBztzt21+CCBztzt.

We deduce

CztY2Cz0Y220tCCBzszs21+CCBzszsds.E19

Using (17) and Schwartz inequality, we get

ztStz0Bz0T0tCCBzszs21+CCBzszsds12,t0T.E20

Since Bis bounded and Ccontinuous, we have

CCBSsz0Ssz02KBzsSsz0z0+CCBzszs,E21

where Kis a positive constant.

Replacing z0by ztin (20) and (21), we get

CCBSsztSszt2KB2z02Ttt+TCCBzszs21+CCBzszsds12+CCBzt+szt+s,ts0.

Integrating this relation over 0Tand using Cauchy-Schwartz, we deduce

0TCCBSsztSsztds2KB2T32+T(1+KBz02×tt+TCCBzszs21+CCBzszsds12,

which achieves the proof.

The following result gives sufficient conditions for strong stabilization of the output (5).

Theorem 1.4 Let Agenerate a semigroup Stt0of contractions on H, Bis a bounded linear operator. If the assumptions 1, 2 of Theorem 1.3 and

0TCCBStψStψdtγY2,ψH,forsomeTγ>0,E22

holds, then control (14) strongly stabilizes the output (5) with decay estimate

CztY=O1t,ast+.E23

Proof: Using (19), we deduce

CzkTY2Czk+1TY22kTkT+1CCBztzt21+CCBztztdt,k0.

From (15) and (22), we have

CzkTY2Czk+1TY2βCzkTY4,E24

where β=γ222KB2T32+T1+KBz022.

Taking sk=CzkTY2, the inequality (24) can be written as

βsk2+sk+1sk,k0.

Since sk+1sk, we obtain

βsk+12+sk+1sk,k0.

Taking ps=βs2and qs=sI+p1sin Lemma 3.3, page 531 in [11], we deduce

skxk,k0,

where xtis the solution of equation xt+qxt=0,x0=s0.

Since xkskand xtdecreases give xt0, t0. Furthermore, it is easy to see that qsis an increasing function such that

0qsps,s0.

We obtain βxt2xt0, which implies that

xt=Ot1,ast+.

Finally the inequality skxk, together with the fact that CztYdecreases, we deduce the estimate (23).

Example 2 Let us consider a system defined on Ω=]0,1[by

zxtt=Azxt+vtaxzxtΩ×]0,+[zx0=z0xΩz0t=z1t=0t>0,E25

where H=L2Ω, Az=z, and aL0,1such that ax0a.e on ]0,1[and axc>0on subregion ωof Ωand v.L0+the control function. System (25) is augmented with the output

wt=χωzt.E26

The operator Agenerates a semigroup of contractions on L2Ωgiven by Stz0=etz0. For z0L2Ωand T=2, we obtain

02χωχωBStz0Stz0dt=02e2tdtωaxz02dxβχωz0L2ω2,

with β=c02e2tdt>0.

Applying Theorem 1.4, we conclude that the control

vt=ωaxz(xt)2dx1+ωaxz(xt)2dx

strongly stabilizes the output (26) with decay estimate

χωztL2ω=O1t,ast+.

2.3 Weak stabilization

The following result provides sufficient conditions for weak stabilization of the output (5).

Theorem 1.5 Let Agenerate a semigroup Stt0of contractions on Hand Bis a compact operator. If the conditions:

  1. ReCCAψψ0,ψDA,

  2. ReCCBψψψ0,ψH,

  3. CCBStψStψ=0,t0=0hold, then control (14) weakly stabilizes the output (5).

Proof: Let us consider the nonlinear semigroup Γtz0ztand let tnbe a sequence of real numbers such that tn+as n+.

From (18), Γtnz0is bounded in H, then there exists a subsequence tϕnof tnsuch that

Γtϕnz0ψ,asn.

Since Bis compact and Ccontinuous, we have

limn+CCBStΓtϕnz0StΓtϕnz0=CCBStψStψ.

For all n, we set

Λntϕnϕn+tCCBΓsz0Γsz021+CCBΓsz0Γsz0ds.

It follows that t0, Λnt0as n+.

Using (15), we get

limn+0tCCBSsΓtϕnz0SsΓtϕnz0ds=0.

Hence, by the dominated convergence Theorem, we have

0tCCBSsψSsψds=0.

We conclude that

CCBSsψSsψ=0,s0t.

Using condition 3 of Theorem 1.5, we deduce that

CΓtϕnz00,asn+.E27

On the other hand, it is clear that (27) holds for each subsequence tϕnof tnsuch that CΓtϕnz0weakly converges in Y. This implies that φY, we have CΓtnz0φ0as n+and hence

CΓtz00,ast+.

Example 3 Consider a system defined in Ω=]0,+[, and described by

zxtt=zxtx+vtBzxtxΩ,t>0zx0=z0xxΩz0t=zt=0t>0,E28

where Az=zxwith domain DA=zH1Ωz0=0zx0asx+and Bz.=01zxdx.is the control operator. The operator Agenerates a semigroup of contractions

Stz0x=z0xtifxt0ifx<t.

Let ω=]0,1[be a subregion of Ωand system (28) is augmented with the output

wt=χωzt.E29

We have

χωχωAzz=01zxzxdx=z2120,

so, the assumption 1 of Theorem 1.5 holds. The operator Bis compact and verifies

χωχωBStz0Stz0=01tz0xdx2,0t1.

Thus

χωχωBStz0Stz0=0,t0z0x=0,a.eonω.

Then, the control

vt=01zxtdx21+01zxtdx2,E30

weakly stabilizes the output (29).

3. Stabilization for semilinear systems

In this section, we give sufficient conditions for exponential, strong and weak stabilization of the output (5). Consider the semilinear system (1) augmented with the output (5).

4. Exponential stabilization

In this section, we develop sufficient conditions for exponential stabilization of the output (5).

The following result concerns the exponential stabilization of (5).

Theorem 1.6 Let Agenerate a semigroup Stt0of contractions on Hand Bbe locally Lipschitz. If the conditions:

  1. ReCCAyy0,yDA,

  2. ReCCByyByy0,yH,

  3. there exist T,γ>0, such that

0TCCBStyStydtγCyY2,yH,E31

hold, then the control

vt=CCBztztzt2,ifzt=0,0,ifzt=0,E32

exponentially stabilizes the output (5).

Proof: Since Stt0is a semigroup of contractions, we have

ddtzt22RevtBztzt.

Integrating this inequality, and using hypothesis 2 of Theorem 1.6, it follows that

ztz0.E33

For all z0Hand t0, we have

CCBStz0Stz0=CCBztztCCBztztStz0+CCBStz0CCBztStz0.

Since Bis locally Lipschitz, there exists a constant positive Lthat depends on z0such that

CCBStz0Stz0CCBztzt+2αLztStz0z0,E34

where αis a positive constant.

Using (33), we deduce

CCBztztvztztz0,t0T.E35

While from the variation of constants formula and using Schwartz’s inequality, we obtain

ztStz0LT0Tvzt2zt2dt12.E36

Integrating (34) over the interval 0Tand taking into account (35) and (36), we get

0TCCBStz0Stz0dt2αT32L2z00Tvzt2zt2dt12+T12z00Tvzt2zt2dt12.

Now, let us consider the nonlinear semigroup Utz0zt[1].

Replacing z0by Utz0in (37), and using the superposition properties of the semigroup Utt0, we deduce that

0TCCBSsUtz0SsUtz0ds2αT32L2Utz0×tt+TvUsz02Usz02ds12+T12Utz0tt+TvUsz02Usz02ds12E37

Thus, by using (31) and (37), it follows that

γCUtz0YMtt+TvUsz02Usz02ds12,E38

where M=2αTL2+1T12is a non-negative constant depending on z0and T.

From hypothesis 1 of Theorem 1.6, we have

ddtCUtz0Y22vUtz02Utz02.E39

Integrating (39) from nTand n+1T,nN, we obtain

CUnTz0Y2CUn+1Tz0Y22nTn+1TvUsz02Usz02ds.

Using (38), (39) and the fact that CUtz0Ydecreases, it follows

1+2γM2CUn+1Tz0Y2CUnTz0Y2.

Then

CUn+1Tz0YβCUnTz0Y,

where β=11+2γM212.

By recurrence, we show that CUnTz0YβnCz0Y.

Taking n=EtTthe integer part of tT, we deduce that

CUtz0YReσtz0,

where R=α1+2γM212, with α>0and σ=ln1+2γM22T>0, which achieves the proof.

4.1 Strong stabilization

The following result provides sufficient conditions for strong stabilization of the output (5).

Theorem 1.7 Let Agenerate a semigroup Stt0of contractions on Hand Bbe locally Lipschitz. If the conditions:

  1. ReCCAyy0,yDA,

  2. ReCCByyByy0,yH,

  3. there exist T,γ>0, such that

0TCCBStyStydtγCyY2,yH,E40

hold, then the control

vt=CCBztzt,E41

strongly stabilizes the output (5).

Proof: From hypothesis 1 of Theorem 1.7, we obtain

ddtCztY22CCBztzt2.E42

Integrating this inequality, gives

20tCCBzszs2dsCz0Y2.

Thus

0+CCBzszs2ds<+,E43

From the variation of constants formula and using Schwartz’s inequality, we deduce

ztStz0LT120TCCBzszs2ds12.E44

Integrating (34) over the interval 0Tand taking into account (44), we obtain

0TCCBSsz0Ssz0ds2αL2T32z020TCCBzszs2ds12+T120TCCBzszs2ds12.

Replacing z0by ztand using the superposition property of the solution, we get

0TCCBSsztSsztds2αL2T32z02tt+TCCBzszs2ds12+T12tt+TCCBzszs2ds12.E45

By (43), we get

tt+TCCBSsztSsztds0,ast+.E46

From (40) and (46), we deduce that CztY0, as t+, which completes the proof.

Proposition 1.8 Let Agenerate a semigroup Stt0of contractions on H, Bbe locally Lipschitz and the assumptions 1, 2 and 3 of Theorem 1.7 hold, then the control (41) strongly stabilizes the output (5) with decay estimate

CztY=Ot12,ast+.E47

Proof: Using (45), we get

0TCCBSsUtz0SsUtz0dsθξt,E48

where θ=2αTL2z02+1T12and ξt=tt+TCCBUsz0Usz02ds.

From (40) and (48), we deduce that

ϱξnTCUnTz0Y2,n0,E49

where ϱ=1γθ.

Integrating the above inequality gives

ddtCUtz0Y22CCBUtz0Utz02,

from nTto n+1T, nNand using (49), we obtain

CUnTz0Y2CUnT+Tz0Y22ξnT,n0.

We obtain

ϱ2CUnT+Tz0Y2ϱ2CUnTz0Y22CUnTz0Y4,n0.E50

Let us introduce the sequence rn=CUnTz0Y2,n0.

Using (50), we deduce that

rnrn+1rn22ϱ2,n0.

Since the sequence rndecreases, we get

rnrn+1rn.rn+12ϱ2,n0,

and also

1rn+11rn2ϱ2,n0.

We deduce that

rnr02r0ϱ2n+1,n0.

Finally, introducing the integer part n=EtTand from (42), the function tCUtz0Ydecreases. We deduce the estimate

CztY=Ot1/2,ast+.

4.2 Weak stabilization

The following result discusses the weak stabilization of the output (5).

Theorem 1.9 Let Agenerate a semigroup Stt0of contractions on H, Bbe locally Lipschitz and weakly sequentially continuous. If assumptions 1, 2 of Theorem 1.7 and

CCBStySty=0,t0Cy=0,E51

hold, then the control

vt=CCBztzt,E52

weakly stabilizes the output (5).

Proof: Let us consider ψYand tn0be a sequence of real numbers such that tn+, as n+.

Using (42), we deduce that the sequence hn=CztnψYis bounded.

Let hγnbe an arbitrary convergent subsequence of hn.

From (33), the subsequence ztγnis bounded in H, so we can extract a subsequence still denoted by ztγnsuch that ztγnφH, as n+.

Since Cis continuous, Bis weakly sequentially continuous and Stis continuous t0, we get

limn+CCBStztγnStztγn=CCBStφStφ.

From (46), we have

0TCCBSsztγnSsztγnds0,asn+.

Using the dominated convergence Theorem, we deduce that

CCBStφStφ=0,forallt0,

which implies, according to (51), that =0, and hence hn0, as t+.

We deduce that CztψY0, as t+. In other words Czt0, as t+, which achieves the proof.

Example 4 Let us consider the system defined in Ω=]0,+[by

zxtt=zxtx+vtBzxt,xΩ,t>0,zx0=z0x,xΩ,E53

where H=L2Ω, Az=zxwith domain DA=zH1Ωz0=0zx0asx+, Bz=01zxdxthe control operator and v.L20+. The operator Agenerates a semigroup of contractions

Stz0x=z0xt,ifxt,0,ifx<t.

Let ω=]0,1[be a subregion of Ωand system (53) is augmented with the output

wt=χωzt.E54

The operator Bis sequentially continuous and verifies

χωχωBStz0Stz0=01tz0xdx01tz0xdx,0t1.

Thus

χωχωBStz0Stz0=0,t0z0x=0a.ex]0,1[,i.eχ]0,1[z0=0.

Then, the control

vt=01zxtdx01zxtdx,E55

weakly stabilizes the output (54).

5. Simulations

Consider system (53) with zx0=sinπx, and augmented with the output (54).

For ω=]0,2[, we have

Figure 1 shows that the output (54) is weakly stabilized on ωwith error equals 6.8×104and the evolution of control is given by Figure 2.

Figure 1.

The stabilization on ω=]0,2[.

Figure 2.

The evolution control in the interval ]0,8].

For ω=]0,3[, we have

Figure 3 shows that the output (54) is weakly stabilized on ωwith error equals 9.88×104and the evolution of control is given by Figure 4.

Figure 3.

The stabilization on ω=]0,3[.

Remark 2. It is clear that the control (55) stabilizes the state on ω, but do not take into account the residual part Ω\ω.

Figure 4.

The evolution control in the interval ]0,12].

6. Conclusions

In this work, we discuss the question of output stabilization for a class of semilinear systems. Under sufficient conditions, we obtain controls depending on the output operator that strongly and weakly stabilizes the output of such systems. This work gives an opening to others questions; this is the case of output stabilization for hyperbolic semilinear systems. This will be the purpose of a future research paper.

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El Hassan Zerrik and Abderrahman Ait Aadi (November 27th 2019). On the Stabilization of Infinite Dimensional Semilinear Systems, Nonlinear Systems -Theoretical Aspects and Recent Applications, Walter Legnani and Terry E. Moschandreou, IntechOpen, DOI: 10.5772/intechopen.87067. Available from:

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