# Approximate Controllability of Semilinear Impulsive Evolution Equations

Authors: Hugo Leiva
Journal: Abstract and Applied Analysis
 (2015)

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## Abstract

We prove the approximate controllability of the following semilinear impulsive evolution equation:z'=Az+Bu(t)+F(t,z,u),z∈Z,t∈(0,τ],z(0)=z0,z(tk+)=z(tk-)+Ik(tk,z(tk),u(tk)),k=1,2,3,…,p, where 0<t1<t2<t3<⋯<tp<τ, Z and U are Hilbert spaces, u∈L2(0,τ;U), B:U→Z is a bounded linear operator, Ik,F:[0,τ]×Z×U→Z are smooth functions, and A:D(A)⊂Z→Z is an unbounded linear operator in Z which generates a strongly continuous semigroup {T(t)}t≥0⊂Z. We suppose that F is bounded and the linear system is approximately controllable on [0,δ] for all δ∈(0,τ). Under these conditions, we prove the following statement: the semilinear impulsive evolution equation is approximately controllable on [0,τ].


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## Body

## 1. Introduction

There are many practical examples of impulsive control systems: a chemical reactor system with the quantities of different chemicals serving as the states variable, a financial system with two state variables of the amount of money in a market and the saving rates of a central bank, and the growth of a population diffusing throughout its habitat which is often modeled by reaction-diffusion equation, for which much has been done under the assumption that the system parameters related to the population environment either are constant or change continuously. However, one may easily visualize situations in nature where abrupt changes such as harvesting, disasters, and instantaneous stoking may occur.This observation motivates us to study the approximate controllability of the following semilinear impulsive evolution equation:(1)z′=Az+Bu(t)+F(t,z,u),z∈Z,t∈(0,τ],z(0)=z0,ztk+=ztk-+Iktk,ztk,utk,k=1,2,3,…,p,where 0<t1<t2<t3<⋯<tp<τ, Z and U are Hilbert spaces, u∈L2(0,τ;U), B:U→Z is a bounded linear operator, Ik,F:[0,τ]×Z×U→Z are smooth functions, and A:D(A)⊂Z→Z is an unbounded linear operator in Z which generates a strongly continuous semigroup {T(t)}t≥0⊂Z.Definition 1 (approximate controllability).
System (1) is said to be approximately controllable on [0,τ] if for every z0, z1∈Z,ε>0 there exists u∈L2(0,τ;U) such that the solution z(t) of (1) corresponding to u verifies (see Figures 1 and 2) (2)z0=z0,z(τ)-z1Z<ε.Figure 1Figure 2We assume the following main hypotheses:(A)
F is a bounded function;(B)
linear system without impulses (8) is approximately controllable on [τ-δ,τ] for all 0<δ<τ.That is, the Gramian controllability operator(3)Qδ=GδGδ∗=∫0δ‍T(t)BB∗T∗(t)dtsatisfies Qδ>0 for all 0<δ<τ, which is equivalent, according to (13) and Lemma 3(c), to the approximate controllability of linear system (8) on [τ-δ,τ], for all 0<δ<τ.This paper has been motivated by the works done in Bashirov and Ghahramanlou [1], Bashirov and Jneid [2], and Bashirov et al. [3], where a new technique to prove the controllability of evolution equations without impulses is used avoiding fixed point theorems, and the work done in [4].The controllability of impulsive evolution equations has been studied recently by several authors, but most of them study the exact controllability only; it is worth mentioning that Chalishajar [5] studied the exact controllability of impulsive partial neutral functional differential equations with infinite delay, Radhakrishnan and Balachandran [6] studied the exact controllability of semilinear impulsive integrodifferential evolution systems with nonlocal conditions, and Selvi and Mallika Arjunan [7] studied the exact controllability for impulsive differential systems with finite delay. To our knowledge, there are a few works on approximate controllability of impulsive semilinear evolution equations worth mentioning: Chen and Li [8] studied the approximate controllability of impulsive differential equations with nonlocal conditions, using measure of noncompactness and Monch fixed point theorem and assuming that the nonlinear term f(t,z) does not depend on the control variable, and Leiva and Merentes in [4] studied the approximate controllability of the semilinear impulsive heat equation using the fact that the semigroup generated by Δ is compact.In this paper, we are not assuming the compactness of the semigroup{T(t)}t≥0 generated by the unbounded operator A; when this semigroup is compact we can consider weaker condition on the nonlinear perturbation F and in the linear part of the system without impulses. Specifically, we can assume the following hypotheses:(a)
(4)Ft,z,uZ≤a0zZα0+b0uZβ0+c0,Ik(t,z,u)Z≤akzZαk+bkuZβk+ck,ggggggggggggggggggk=1,2,3,…,p,with1/2≤αk<1, 1/2≤βk<1, k=0,1,2,3,…,p;(b)
the linear system is approximately controllable only on[0,τ].This case is similar to the semilinear impulsive heat equations studied in [4], where the authors use conditions (a) and (b), the compactness of the semigroup generated by the Laplacian operator Δ, and Rothe’s fixed point theorem to prove the approximate controllability of the system on [0,τ].When it comes to the wave equation, the situation is totally different; the semigroup generated by the linear part is not compact; it is in fact a group, which can never be compact. Furthermore, if the control acts on a portionω of the domain Ω for the spatial variable, then the system is approximately controllable only on [0,τ] for τ≥2, which was proved in [9], where the following system governed by the wave equations was studied: (5)ytt=Δy+1ωu(t,x),on⁡(0,τ)×Ω,y=0,on⁡(0,τ)×∂Ω,y(0,x)=y0(x),yt(0,x)=y1(x),inΩ,where Ω is a bounded domain in Rn, ω is an open nonempty subset of Ω, 1ω denotes the characteristic function of the set ω, the distributed control u∈L2([0,τ];L2(Ω)), and y0∈H2(Ω)∩H01, y1∈L2(Ω).However, if the control acts on the whole domainΩ, it was proved in [10] that the system is controllable [0,τ], for all τ>0. More specifically, the authors studied the following system: (6)ytt=Δy+u(t,x),on⁡(0,τ)×Ω,y=0,on⁡(0,τ)×∂Ω,y(0,x)=y0(x),yt(0,x)=y1(x),inΩ,where Ω is a bounded domain in Rn, the distributed control u∈L2([0,τ];L2(Ω)), and y0∈H2(Ω)∩H01, y1∈L2(Ω).To justify the use of this new technique [1], in this paper, we consider as an application the semilinear impulsive wave equation with controls acting on the whole domain Ω, so that hypotheses (A) and (B) hold: (7)ytt=Δy+u(t,x)+f(t,y,yt,u(t)),on⁡0,τ×Ω,y=0,on⁡(0,τ)×∂Ω,y(0,x)=y0(x),yt(0,x)=y1(x),inΩ,yttk+,x=yt(tk-,x)+Ikt,ytk,x,yttk,x,utk,x,x∈Ω,where 0<t1<t2<t3<⋯<tp<τ, Ω is a bounded domain in Rn, the distributed control u∈L2([0,τ];L2(Ω)), y0∈H2(Ω)∩H01, y1∈L2(Ω), and Ik,f are smooth functions with f beingbounded.


## 2. Controllability of the Linear Equation without Impulses

In this section we will present some characterization of the approximate controllability of the corresponding linear equations without impulses. To this end, we note that for allz0∈Z and u∈L2(0,τ;U) the initial value problem (8)z′=Az+Bu(t),z∈Z,z(t0)=z0admits only one mild solution given by (9)zt=z(t,t0,z0,u)=T(t)z0+∫t0t‍T(t-s)Bu(s)ds,t∈t0,τ,0≤t0≤τ.Definition 2.
For system (8), one defines the following concept: the controllability maps Gτδ:L2τ-δ,τ;U→Z, Gδ:L2(0,δ;U)→Z, defined by (10)Gτδu=∫τ-δτ‍T(τ-s)Bu(s)ds,u∈L2(τ-δ,τ;U),Gδv=∫0δ‍T(s)Bv(s)ds,v∈L20,δ;U,satisfy the following relation: (11)Gτδu=∫τ-δτ‍T(τ-s)Bu(s)ds=∫0δ‍T(s)Bu(τ-s)ds=Gδu(τ-·).The adjoints of these operators Gτδ∗:Z→L2(τ-δ,τ;U), Gδ∗:Z→L2(0,δ;U) are given by (12)Gτδ∗zt=B∗T∗τ-tz,t∈τ-δ,τ,(Gδ∗x)(t)=B∗T∗(t)z,t∈[0,δ].The Gramian controllability operators are given by (3) and (13)Qτδ=GτδGτδ∗=∫τ-δτ‍T(τ-t)BB∗T∗(τ-t)dt=Qδ.The following lemma holds in general for a linear bounded operatorG:W→Z between Hilbert spaces W and Z (see [4, 11, 12]).Lemma 3.
The following statements are equivalent to the approximate controllability of the linear system (8) on [τ-δ,τ]. (a)
Range(Gτδ)¯=Z.(b)
Ker⁡(Gτδ∗)={0}.(c)
〈Qτδz,z〉>0, z≠0 in Z.(d)
limα→0+α(αI+Qτδ)-1z=0.(e)
For allz∈Z, one has Gτδuα=z-α(αI+QTδ)-1z, where (14)uα=Gτδ∗αI+Qτδ-1z,α∈(0,1].Solimα→0Gτδuα=z and the error Eτδz of this approximation is given by the formula (15)Eτδz=ααI+Qτδ-1z,α∈(0,1].(f)
Moreover, if one considers for eachv∈L2τ-δ,τ;U the sequence of controls given by (16)uα=Gτδ∗αI+Qτδ-1z+v-Gτδ∗αI+Qτδ-1Gτδv,α∈0,1,one gets that(17)Gτδuα=z-ααI+QTδ-1z+Gτδv,limα→0Gτδuα=z,with the errorEτδz of this approximation given by the formula (18)Eτδz=ααI+Qτδ-1z+Gτδv,α∈(0,1].Remark 4.
The foregoing lemma implies that the family of linear operatorsΓατδ:Z→W, defined for 0<α≤1 by (19)Γατδz=Gτδ∗αI+Qτδ-1z,is an approximate inverse for the right of the operator W, in the sense that (20)limα→0GτδΓατδ=Iin the strong topology.Lemma 5.
Q
δ
>
0 if and only if linear system (8) is controllable on [τ-δ,τ]. Moreover, given an initial state y0 and a final state z1, one can find a sequence of controls {uαδ}0<α≤1⊂L2(τ-δ,τ;U): (21)uα=Gτδ∗αI+GτδGτδ∗-1z1-Tτy0,α∈0,1,such that the solutions y(t)=y(t,τ-δ,y0,uαδ) of the initial value problem (22)y′=Ay+Buα(t),y∈Z,t>0,y(τ-δ)=y0satisfy (23)limα→0+yτ,τ-δ,y0,uα=z1;that is, (24)limα→0+y(τ)=limα→0+T(δ)y0+∫τ-δτ‍Tτ-sBuα(s)ds=z1.


## 3. Controllability of the Semilinear Impulsive System

In this section, we will prove the main result of this paper: the approximate controllability of the semilinear impulsive evolution equation given by (1). To this end, for all z0∈Z and u∈L2(0,τ;U), the initial value problem (25)z′=Az+Bu+F(t,z,u),t∈(0,τ],t≠tk,z∈Z,z(0)=z0,ztk+=ztk-+Ikt,ztk,utk,k=1,2,3,…,p,admits only one mild solution given by(26)zut=T(t)z0+∫0t‍T(t-s)Bu(s)ds+∫0t‍Tt-sFs,zus,usds+∑0<tk<t‍Tt-tkIktk,ztk,utk,t∈[0,τ].Now, we are ready to present and prove the main result of this paper, which is the approximate controllability of semilinear impulsive equation (1).Theorem 6.
Under conditions (A) and (B), semilinear impulsive system (1) is approximately controllable on [0,τ].Proof.
Given an initial statez0, a final state z1, and ϵ>0, we want to find a control uαδ∈L2(0,τ;U) steering the system from z0 to an ϵ-neighborhood of z1 on time τ. Specifically, (27)limα→0∑0<tk<τ‍T(τ)z0+∫0τ‍T(τ-s)Buαδ(s)ds+∫0τ‍Tτ-sFs,zαs,uαδsds+∑0<tk<τ‍Tτ-tkIkzαtk,uαδtk=z1.Consider any u∈L2(0,τ;U) and the corresponding solution z(t)=z(t,0,z0,u) of initial value problem (25). For α∈(0,1], we define the control uαδ∈L2(0,τ;U) as follows: (28)uαδ(t)=u(t),if0≤t≤τ-δ,uα(t),ifτ-δ<t≤τ,where (29)uαt=B∗T∗τ-tαI+GτδGτδ∗-1z1-Tδzτ-δ,τ-δ<t≤τ.Now, assume that 0<δ<τ-tp. Then the corresponding solution zαδ(t)=z(t,0,z0,uαδ) of initial value problem (25) at time τ can be written as follows: (30)zαδτ=T(τ)z0+∫0τ‍Tτ-sBuαδ(s)ds+∫0τ‍Tτ-sFs,zαδs,uαδsds+∑0<tk<τ‍Tτ-tkIkezαδtk,uαδtk=T(δ)∑0<tk<τ-δ‍Tτ-δz0+∫0τ-δ‍Tτ-δ-sBuαδ(s)ds+∫0τ-δ‍Tτ-δ-sFs,zαδs,uαδsds+∑0<tk<τ-δ‍Tτ-δ-tkIkezαδtk,uαδtk+∫τ-δτ‍T(τ-s)Buαδ(s)ds+∫τ-δτ‍T(τ-s)F(s,zαδ(s),uαδ(s))ds=T(δ)z(τ-δ)+∫τ-δτ‍T(τ-s)Buα(s)ds+∫τ-δτ‍Tτ-sFs,zαδs,uαsds.So, (31)zαδτ=T(δ)z(τ-δ)+∫τ-δτ‍T(τ-s)Buα(s)ds+∫τ-δτ‍Tτ-sFs,zαδs,uαsds.The corresponding solution yαδ(t)=y(t,τ-δ,z(τ-δ),uα) of initial value problem (22) at time τ is given by (32)yαδ(τ)=T(δ)z(τ-δ)+∫τ-δτ‍T(τ-s)Buα(s)ds.Therefore, (33)zαδ(τ)-yαδ(τ)≤∫τ-δτ‍T(τ-s)F(s,zαδ(s),uα(s))ds.Then, putting M=sup⁡0≤t≤τT(t) and K=sup⁡[0,τ]×Z×UF(t,z,u), we obtain the following estimate: (34)zαδ(τ)-yαδ(τ)≤MKδ.Hence, (35)zαδ(τ)-z1≤MKδ+yαδ(τ)-z1.From Lemma 5, there exists α>0 such that (36)yαδ(τ)-z1≤ϵ2.So, taking δ<min⁡{τ-tp,ϵ/2MK}, we obtain for the corresponding control uαδ that (37)zαδ(τ)-z1≤ϵ.Geometrically, the proof goes as shown in Figure 3.
This completes the proof of the theorem.Figure 3


## 4. Applications

As an application, we will prove the approximate controllability of the following control system governed by the semilinear impulsive wave equation:(38)ytt=Δy+u(t,x)+f(t,y,yt,u(t)),on⁡(0,τ)×Ω,y=0,on⁡(0,τ)×∂Ω,y(0,x)=y0(x),yt(0,x)=y1(x),inΩ,yttk+,x=yttk-,x+Ikt,ytk,x,yttk,x,utk,x,x∈Ω,where Ω is a bounded domain in Rn, the distributed control u∈L2([0,τ];L2(Ω)), y0∈H2(Ω)∩H01, y1∈L2(Ω), and Ik,f are smooth functions with f beingbounded.


### 4.1. Abstract Formulation of the Problem

In this part, we will choose the space where this problem will be set up as an abstract control system in a Hilbert space. LetX=L2(Ω)=L2(Ω,R) and consider the linear unbounded operator A:D(A)⊂X→X defined by Aϕ=-Δϕ, where (39)D(A)=H2(Ω,R)∩H01(Ω,R).Then the eigenvalues λj of A have finite multiplicity γj equal to the dimension of the corresponding eigenspace and 0<λ1<λ2<⋯<λn→∞. Moreover, consider the following.(a) There exists a complete orthonormal set{ϕj,k} of eigenvectors of A.(b) For allx∈D(A), we have(40)Ax=∑j=1∞‍λj∑k=1γj‍x,ϕj,kϕj,k=∑j=1∞‍λjEjx,where·,· is the usual inner product in L2 and(41)Ejx=∑k=1γj‍x,ϕj,kϕj,k,which means the set{Ej}j=1∞ is a complete family of orthogonal projections in X and x=∑j=1∞Ejx, x∈X.(c)-A generates an analytic semigroup {e-At} given by(42)e-Atx=∑j=1∞‍e-λjtEjx.(d) The fractional powered spacesXr are given by (43)Xr=DAr=x∈X:∑n=1∞‍λn2rEnx2<∞,r≥0,with the norm(44)xr=Arx=∑n=1∞‍λn2rEnx21/2,x∈Xr,Arx=∑n=1∞‍λnrEnx.Also, forr≥0, we define Zr=Xr×X, which is a Hilbert space endowed with the norm given by (45)yvZr=yr2+v2.Then, (1) can be written as abstract second order ordinary differential equations in Z1/2 as follows: (46)y′′=-Ay+u+fe(t,y,y′,u),t∈(0,τ],t≠tk,y(0)=y0,y′(0)=y1,y′tk+=y′tk-+Iketk,ytk,y′tk,utk,k=1,2,3,…,p,where Ike,fe:[0,τ]×Z1/2×U→Z1/2 are defined by (47)Iket,y,v,ux=Ikt,yx,vx,ux,fet,z,ux=ft,yx,vx,ux,∀x∈Ω,k=1,2,…,p.With the change of variables y′=v, we can write second order equation (46) as a first order system of ordinary differential equations in the Hilbert space Z1/2=X1/2×X as follows: (48)z′=Az+Bu+F(t,z,u(t)),z∈Z1/2,t∈(0,τ],t≠tk,z(0)=z0,z(tk+)=z(tk-)+Jktk,ztk,utk,k=1,2,3,…,p,where u∈L2([0,τ];U), U=X=L2(Ω), (49)z=yv,B=0I,A=0IX-A0is an unbounded linear operator with domain D(A)=D(A)×D(A1/2), and Jk,F:[0,τ]×Z1/2×U→Z1/2 are defined by (50)Ft,z,u=0fet,y,v,u,Jk(t,z,u)=0Ike(t,w,v,u).It is well known that the operatorA generates a strongly continuous group T(t)t≥0 in the space Z=Z1/2=X1/2×X (see [13]). The following representation for this group can be found in [9] as Theorem 2.2.Proposition 7.
The groupT(t)t≥0 generated by the operator A has the following representation: (51)T(t)z=∑nj=1∞‍eAjtPjz,z∈Z1/2,t≥0,where Pjj≥0 is a complete family of orthogonal projections in the Hilbert space Z1/2 given by (52)Pj=Ej00Ej,j≥1,Aj=RjPj,Rj=01-λj0,j≥1.


### 4.2. Approximate Controllability

Now, we are ready to formulate and prove the main result of this section, which is the approximate of the semilinear impulsive wave equation withbounded nonlinear perturbation.Theorem 8.
Semilinear impulsive wave equation (38) is approximately controllable on [0,τ].Proof.
From [10], we know that the corresponding linear system without impulses (53)z′=Az+Bu,z∈Z1/2,t∈(0,τ],z(0)=z0is exactly controllable on [0,δ], for all 0<δ<τ. On the other hand, since f is bounded, there exists M>0 such that sup⁡[0,τ]×R×R×Rf(t,y,v,u)≤M. Therefore, we obtain(54)F(t,z,u)Z1/2=∫Ω‍ft,yx,vx,ux2dx1/2≤μ(Ω)M,for all (t,z,u)∈[0,τ]×Z1/2×U with μ(Ω) the Lebesgue measure of Ω. Hence hypotheses (A) and (B) in Theorem 6 are satisfied and we get the result.


## 4.1. Abstract Formulation of the Problem

In this part, we will choose the space where this problem will be set up as an abstract control system in a Hilbert space. LetX=L2(Ω)=L2(Ω,R) and consider the linear unbounded operator A:D(A)⊂X→X defined by Aϕ=-Δϕ, where (39)D(A)=H2(Ω,R)∩H01(Ω,R).Then the eigenvalues λj of A have finite multiplicity γj equal to the dimension of the corresponding eigenspace and 0<λ1<λ2<⋯<λn→∞. Moreover, consider the following.(a) There exists a complete orthonormal set{ϕj,k} of eigenvectors of A.(b) For allx∈D(A), we have(40)Ax=∑j=1∞‍λj∑k=1γj‍x,ϕj,kϕj,k=∑j=1∞‍λjEjx,where·,· is the usual inner product in L2 and(41)Ejx=∑k=1γj‍x,ϕj,kϕj,k,which means the set{Ej}j=1∞ is a complete family of orthogonal projections in X and x=∑j=1∞Ejx, x∈X.(c)-A generates an analytic semigroup {e-At} given by(42)e-Atx=∑j=1∞‍e-λjtEjx.(d) The fractional powered spacesXr are given by (43)Xr=DAr=x∈X:∑n=1∞‍λn2rEnx2<∞,r≥0,with the norm(44)xr=Arx=∑n=1∞‍λn2rEnx21/2,x∈Xr,Arx=∑n=1∞‍λnrEnx.Also, forr≥0, we define Zr=Xr×X, which is a Hilbert space endowed with the norm given by (45)yvZr=yr2+v2.Then, (1) can be written as abstract second order ordinary differential equations in Z1/2 as follows: (46)y′′=-Ay+u+fe(t,y,y′,u),t∈(0,τ],t≠tk,y(0)=y0,y′(0)=y1,y′tk+=y′tk-+Iketk,ytk,y′tk,utk,k=1,2,3,…,p,where Ike,fe:[0,τ]×Z1/2×U→Z1/2 are defined by (47)Iket,y,v,ux=Ikt,yx,vx,ux,fet,z,ux=ft,yx,vx,ux,∀x∈Ω,k=1,2,…,p.With the change of variables y′=v, we can write second order equation (46) as a first order system of ordinary differential equations in the Hilbert space Z1/2=X1/2×X as follows: (48)z′=Az+Bu+F(t,z,u(t)),z∈Z1/2,t∈(0,τ],t≠tk,z(0)=z0,z(tk+)=z(tk-)+Jktk,ztk,utk,k=1,2,3,…,p,where u∈L2([0,τ];U), U=X=L2(Ω), (49)z=yv,B=0I,A=0IX-A0is an unbounded linear operator with domain D(A)=D(A)×D(A1/2), and Jk,F:[0,τ]×Z1/2×U→Z1/2 are defined by (50)Ft,z,u=0fet,y,v,u,Jk(t,z,u)=0Ike(t,w,v,u).It is well known that the operatorA generates a strongly continuous group T(t)t≥0 in the space Z=Z1/2=X1/2×X (see [13]). The following representation for this group can be found in [9] as Theorem 2.2.Proposition 7.
The groupT(t)t≥0 generated by the operator A has the following representation: (51)T(t)z=∑nj=1∞‍eAjtPjz,z∈Z1/2,t≥0,where Pjj≥0 is a complete family of orthogonal projections in the Hilbert space Z1/2 given by (52)Pj=Ej00Ej,j≥1,Aj=RjPj,Rj=01-λj0,j≥1.


## 4.2. Approximate Controllability

Now, we are ready to formulate and prove the main result of this section, which is the approximate of the semilinear impulsive wave equation withbounded nonlinear perturbation.Theorem 8.
Semilinear impulsive wave equation (38) is approximately controllable on [0,τ].Proof.
From [10], we know that the corresponding linear system without impulses (53)z′=Az+Bu,z∈Z1/2,t∈(0,τ],z(0)=z0is exactly controllable on [0,δ], for all 0<δ<τ. On the other hand, since f is bounded, there exists M>0 such that sup⁡[0,τ]×R×R×Rf(t,y,v,u)≤M. Therefore, we obtain(54)F(t,z,u)Z1/2=∫Ω‍ft,yx,vx,ux2dx1/2≤μ(Ω)M,for all (t,z,u)∈[0,τ]×Z1/2×U with μ(Ω) the Lebesgue measure of Ω. Hence hypotheses (A) and (B) in Theorem 6 are satisfied and we get the result.


## 5. Final Remark

This technique can be applied to those systems where the linear part does not generate a compact semigroup and is controllable on any[0,δ] for δ>0 and the nonlinear perturbation is bounded. Example of such systems is the following controlled thermoelastic plate equation whose linear part was studied in [10]: (55)ytt+Δ2y+αΔθ=u1t,x+f1t,y,yt,θ,ut,on⁡0,τ×Ω,θt-βΔθ-αΔyt=u2t,x+f2t,y,yt,θ,ut,on⁡0,τ×Ω,θ=y=Δy=0,on⁡(0,τ)×∂Ω,yttk+,x=yttk-,x+Ik1t,ytk,x,yttk,x,utk,x,x∈Ω,θ(tk+,x)=θ(tk-,x)+Ik2tk,θtk,x,utk,x,x∈Ω,in the space Z=X1×X×X, where Ω is a bounded domain in Rn, the distributed controls u1,u2∈L2([0,τ];L2(Ω)), and Iki,fi are smooth functions with fi, i=1,2, beingbounded.



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