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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: 6

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. 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,

  • 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

  • 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.

    Notes

    • These authors contributed equally.

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    El Hassan Zerrik and Abderrahman Ait Aadi (November 27th 2019). On the Stabilization of Infinite Dimensional Semilinear Systems [Online First], IntechOpen, DOI: 10.5772/intechopen.87067. Available from:

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