1. Introduction

In this paper, we study the interior approximate controllability of the following strongly damped semilinear wave equations under the influence of impulses, delays and nonlocal conditions; without using fixed point Theorem

In the space Z1/2=D((-Δ)1/2)×L2(Ω) where w'=∂w∂t, w''=∂2w∂t2, Ω⊂RN, N≥1 is a bounded domain, γ and η are positive numbers.

Along with Dirichlet boundary condition, where Δ denotes de Laplacian operator, Ω is a bounded domain in RN (N≥1), ∂Ω denotes the boundary of Ω, Ωτ=(0,τ]×Ω,  Ω∂=(0,τ)×∂Ω,  Ω-r=[-r,0]×Ω,  ω is an open nonempty subset of Ω, 1ω denotes the characteristic function of the set ω, the distributed control u belongs to L2([0,τ];L2(Ω)), ϕi:[-r,0]×Ω→R, i=1,2, are continuous functions, 0<r1<…<rm<r are the delays and 0<τ1<…<τq<τ.

where ρ:R+→[0,∞) is a continuous and increasing function. In particular, ρ could be given by

ρ(w)=a0wβ+b0, w>0,β>0,a0,>0.

Moreover,

y(tk,x)=y(tk+,x)=limt→tk+y(t,x), y(tk-,x)=limt→tk-y(t,x).

To set this problem, we shall choose the following natural Banach space:

PCt1..tP([-r,τ];Z1/2)={z:J=[0,τ]→Z1/2:z∈C(J';Z1/2),∃z(tk+,⋅),z(tk-,⋅)andz(tk,⋅)=z(tk+,⋅)},

J'=[-r,τ]∖{t1,t2,…,tp} endowed with the norm z=supt∈[-r,τ]z(t,⋅)Z1/2, where z=(w,v)⊤=(w,wt)⊤ and

zZ1/2=∫Ω((-Δ)1/2w2+v2)dx1/2,for allz∈Z1/2.

Remark 1. It is clear that PCt1..tP([-r,τ];Z1/2) is a closed linear subspace of the Banach space of all piecewise continuous functions PC([-r,τ];Z1/2) with the supreme norm, which implies that PCt1..tP([-r,τ];Z1/2) is a Banach space with the same norm.

We note that the interior controllability of the following strongly damped wave equation without impulses, delays and nonlocal conditions

has been proved in , where the abstract formulation is done using fractional powered spaces, some ideas are taken from it to study this present problem. Finally, the approximate controllability of the system (1) follows from the approximate controllability of the linear system (4) in any interval of the form [τ-δ,τ], with 0<δ<τ, and using a new technique avoid fixed point theorems by applying in (^{Bashirov A.E. and Ghahramanlou N. (2013)}), (^{Bashirov et al.(2007)}), (^{Bashirov A.E. and Mahmudov N.I. (1999)}).

There are many practical examples of impulsive control systems, a chemical reactor system, a financial system with two state variables, the amount of money in a market and the savings rate of a central bank; and the growth of a population diffusing throughout its habitat modeled by a reaction-diffusion equation. One may easily visualize situations in these examples where abrupt changes such as harvesting, disasters and instantaneous stocking may occur. These problems are modeled by impulsive differential equations, and for more information see the monographs, (^{Lakshmikantham V., Bainov D. D. and Simeonov P.S. (1989)}) and (^{Samoilenko A. M. and Perestyuk N.A. (1995)}) 

The controllability of Impulsive Evolution Equations has been studied recently for several authors, but most them study the exact controllability only, to mention: (^{Chalishajar D. N. (2011)}), studied the exact controllability of impulsive partial neutral functional differential equations with infinite delay and (^{Selvi S. and Mallika Arjunan M. (2012)}) 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, to mention: (^{Chen L. and Li G (2010)}) studied the Approximate controllability of impulsive differential equations with nonlocal conditions, using measure of noncompactness and Monch’s fixed point theorem, and assuming that the nonlinear term f(t,z) does not depend on the control variable.

Recently, in Leiva Hugo (^{2014 a}, ^b); ^{Leiva Hugo and Merentes N. (2015)}; ^{Leiva Hugo (2015)} the approximate controllability of semilinear evolution equations with impulses has been studied applying Rothe’s Fixed Point Theorem. Also, there are many papers on evolution equations with impulses and delay or with impulses and nonlocal conditions or with local conditions and delays, where not only the controllability is studied, but also other aspects are studied, such as the existence of mild solutions, synchronization, stability, etc. To mention, we have the following references: ^{Chalishajar D. N. (2011)}; ^{Chen L. and Li G (2010)}; ^{Chiu K. and Li T. (2019)}; ^{Guevara C. and Leiva H. (2016)}; ^{Guevara C. and Leiva H. (2017)}; ^{Jiang C., Zhang F. and Li Tongxing (2018)}; ^{Leiva Hugo and Rojas Raul (2016)}; ^{Li Tongxing, Pintus Nicola and Viglialoro Giuseppe (2019)}; ^{Liang Jin, Liu James H. and XiaoTi-Jun (2009)}; ^{Qina Haiyong et al. (2017)}; ^{Quin Haiyong et al. (2017)}; ^{Selvi S. and Mallika Arjunan M. (2012)}.

2. Abstract Formulation of the Problem

In this section, we choose a Hilbert space where system (1) can be written as an abstract differential equation; to this end, we shall use the following notations:

Let X=U=L2(Ω)=L2(Ω,R) and consider the linear unbounded operator :D(A)⊂X→X defined by

Aφ=-Δφ, where DA=H2Ω, R∩H01Ω, R

The fractional powered spaces Xα (see details in ^{Larez H., Leiva Hugo and Rebaza J. (2012)}) are given by

Xα=D(Aα)=x∈X:∑n=1∞λn2αEnx2<∞,

endowed with the norm

xα=Aαx=∑n=1∞λn2αEnx21/2,

where {Ej} is a family of complete orthogonal projections in X; and for the Hilbert sapce Zα=Xα×X the corresponding norm is

wvZα=wα2+v2.

Proposition 1. Given j≥1, the operator Pj:Zα→Zα defined by

is a continuous (bounded) orthogonal projections in the Hilbert space Zα.

Hence, the equation (1) can be written as an abstract second order ordinary differential equation in X as

where

Ike:[0,τ]×Z1/2×U→X(t,w,v,u)(⋅)↦Ik(t,w(⋅),v(⋅),u(⋅)),,

fe:[0,τ]×Z1/2×Cm([-r,0];Z1/2)×Cm([-r,0];Z1/2)×U→X(t,w,v,φ1,…,φm,ψ1,…,ψm,u)(⋅)↦f(t,w(⋅),v(⋅),φ1(-r1,⋅),…φ(-rm,⋅,)ψ1(-r1,⋅),…,ψm(-rm,⋅),u(⋅)),

Bω:U→Uu(⋅)↦1ωu(⋅),,

gi:Cq([-r,0];X)→C([-r,0];Z1/2)gi(φ1,…,φq)(s,⋅)↦hi(φ1(s,⋅),…,φq(s,⋅)),i=1,2.

A change of variable v=w' transforms the second order equation (6) into the following first order system of ordinary differential equations with impulses, delays and nonlocal conditions in the space Z1/2.

where u∈C([0,τ],U), z=(w,v)⊤, ϕ=(ϕ1,ϕ2)⊤∈C([-r,0],X), zt defined as a function from [-r,0] to Z1/2 by zt(s)=z(t+s),-r≤s≤0,

A=0IX-γA-ηA1/2

is an unbounded linear operator with domain D(A)=D(A)×D(A1/2), IX represents the identity in X,

Bω:U→Z1/2u↦(0,Bωu)⊤,

Ik[0,τ]×Z1/2×U→Z1/2(t,z,u)↦(0,Ike(t,w,v,u))⊤,

F:[0,τ]×Z1/2×Cm([-r,0],Z1/2)×U→Z1/2(t,z,ϕ1,…,ϕm,u)↦0fe(t,w,v,ϕ11(-r1),…,ϕ1m(-rm),ϕ21(-r1),…,ϕ2m(-rm),u),

and

g:Cq([-r,0];X)→C([-r,0];X)×C([-r,0];X)g(ϕ1,…,ϕq)(s,⋅)↦g1(ϕ11(s,⋅),…,ϕ1q(s,⋅)g2(ϕ21(s,⋅),…,ϕ2q(s,⋅)

Definition 1. (Approximate Controllability) The system (7) is said to be approximately controllable on [0,τ] if for every ϕ=(ϕ1,ϕ2)∈C([-r,0];Z1/2), z1∈Z1/2 and ε>0, there exists u∈C([0,τ];L2(Ω)) such that the solution z(t) of (1) corresponding to u verifies:

z(0)+h(zτ1,…,zτq)(0)=ϕ(0), and z(τ)-z1Z1/2<ε.

The hypotheses H1) and H2), together with the continuous inclusion X1/2⊂X, yield.

Proposition 2. The function F satisfies the following inequality:

where σ:R+→[0,∞) is a continuous function.

It is well known that the operator A generates a strongly continuous semigroup {T(t)}t≥0 in the space Z1/2, which is also analytic. Furthermore, Lemma 2.1 in ^{Leiva Hugo (2003)} yields.

Proposition 3.

The semigroup {T(t)}t≥0 generated by the operator A is compact and has the following representation

where {Pj}j≥0 is a complete family of orthogonal projections in the Hilbert space Z1/2 given by (5) and

Aj=RjPj, Rj=01-γλj-ηλj1/2, j≥1.

Moreover, eAjt=eRjtPj, the eigenvalues of Rj are

λ=-lj1/2η±η2-4γ2, j=1,2,…,

Aj*=RjPj, Rj*=0-1γλj-ηλj1/2.

and

∥T(t)∥≤M(η,γ)e-βt,  t≥0,

where

β=λ112min⁡Reη±η2-4γ2

3. APPROXIMATE CONTROLLABILITY OF THE LINEAR SYSTEM

In this section, we shall characterize the approximate controllability of the linear system. To this end, for all z0∈Z1 and u∈L2([0,τ];U) the initial value problem

admits only one mild solution given by

For the system (10) we define the following concept: The controllability map (for τ>0) Gτδ:L2([τ-δ,τ];U)→Z1/2 is defined by

whose adjoint operator Gτδ* is

Gτδ*:Z1/2→L2([τ-δ,τ];U)

is given by

The Gramian controllability operator is defined as:

The following lemma holds in general for a linear bounded operator G:W→Z between Hilbert spaces W and Z(see ^{Curtain R.F. and Pritchard A.J. (1978)}, ^{Curtain R.F. and Zwart H.J. (1995)} and ^{Leiva Hugo, Merentes N. and Sanchez J. (2012)}).

The following Theorem is a characterization of the approximate controllability of the system (10):

Theorem 1. (see ^{Bashirov et al. (2007)}, ^{Bashirov A.E. and Mahmudov N.I. (1999)}, ^{Curtain R.F. and Pritchard A.J. (1978)}, ^{Curtain R.F. and Zwart H.J. (1995)} and ^{Leiva Hugo, Merentes N. and Sanchez J. (2012)}) The system (10) is approximately controllable on [0,τ] if, and only if, any one of the following conditions hold:

Remark 2. The Theorem 1 implies that the family of linear operators Γτδ:Z1/2→L2([τ-δ,τ];U) defined for 0≤α≤1, by

Γτδz=Gτδ*(αI+Qτδ*)-1z,

satisfies the following relation

limα→0GτδΓτδ=I

in the strong topology.

Since the controllability of the linear system (10) was prove by ^{Carrasco A., Leiva H. and Sanchez J.L. (2013)}, on [0,τ] for all τ>0, we get the following characterization for the approximate controllability of (10).

Lemma 2. Qτδ>0 if, and only if, the linear system (10) is controllable on [τ-δ,τ]. Moreover, given an initial state y0 and a final state Z1/2 we can find a sequence of controls uαδ}0<α≤1⊂L2(τ-δ,τ;U)

uα=Gτδ*(αI+GτδGτδ*)-1(z1-T(τ)y0), α∈(0,1],

such that the solutions y(t)=y(t,τ-δ,y0,uαδ) of the initial value problem

satisfies

limα→0+y(τ,τ-δ,y0,uα)=z1.

e.i.,

limα→0+y(τ)=limα→0+T(δ)y0+∫τ-δτT(τ-s)Buα(s)ds=z1.

3. The System with Impulses, Delays and Nonlocal Conditions

In this section, we shall prove the main result of this paper, the interior approximate controllability of the semilinear strongly damped wave equation with impulses, delays and nonlocal conditions given by (1), which is equivalent to prove the approximate controllability of the system (7). To this end, for all ϕ∈C and u∈C([0,τ];U) the initial value problem, according with the recent work from ^{Leiva Hugo and Sundar P. (2017)}; ^{Leiva Hugo (2018)}

admits only one mild solution z∈PCt1..tP([-r,τ];Z1/2) given by

z(t)=T(t)ϕ(0)-T(t)[(g(zt1,…,ztq))(0)]+∫0tT(t-s)Bωu(s)ds+∫0tT(t-s)F(s,z(s),zs(-r1),zs(-r2),…,zs(-rm),u(s))ds+∑0<tk<tT(t-tk)Ik(tk,z(tk),u(tk)),  t∈[0,τ],z(t)+(g(zτ1,…,zτq))(t)=ϕ(t), t∈[-r,0].

Now, we are ready to present and prove the main result of this paper, which is the interior approximate controllability of the semilinear strongly damped wave equation with impulses, delays and nonlocal conditions (16).

Theorem 2. If the functions f,Ik,h are smooth enough, condition (1) holds, and since the linear system (10) is approximately controllable on any interval [τ-δ,τ], 0<δ<τ, then system (16) is approximately controllable on [0,τ].

Demostration. Given ϕ∈C, a final state z1 and ε>0, we want to find a control uδ∈L2(0,τ;U) steering the system to z1 on [τ-δ,τ]. Precisely, for 0<δ<min{τ-tp,r} small enough, there exists control uδ∈L2(0,τ;U) such that corresponding of solutions zδ of (16) satisfies

zδ(τ)-z1<ϵ.

In fact, we consider any fixed control u∈L2(0,τ;U) and the corresponding solution z(t)=z(t,0,ϕ,u) of the problem (16). For 0<δ<min{τ-tp,r} small enough, we define the control uδ∈L2(0,τ;U) as follows

uδ(t)=u(t),if0≤t≤τ-δ,vδ(t),ifτ-δ<t≤τ.

where

vδ(t)=Bω*T*(τ-t)(αI+GτδGτδ*)-1(z1-T(δ)z(τ-δ)), τ-δ<t≤τ.

Since 0<δ<τ-tp, then τ-δ>tp, the corresponding solution zδ(t)=z(t,0,ϕ,uδ) of the nonlocal Cauchy problem (16) at time τ can be written as follows:

zδ(τ)=T(τ)ϕ(0)-T(τ)[(g(zτ1,…,zτq)(0)]+∫0τT(τ-s)B(s)uδ(s)ds+∫0τT(τ-s)F(s,zδ(s),zsδ(-r1),zsδ(-r2),…,zsδ(-rm),u(s))+∑0<tk<τT(τ-tk)Ik(tk,z(tk),uδ(tk))=T(δ){T(τ-δ)ϕ(0)-T(τ-δ)[(g(zτ1,…,zτ1))(0)]+∫0τ-δT(τ-δ-s)Bω(s)(s)uδ(s)ds+∫0τ-δT(τ-δ-s)F(s,zδ(s),zsδ(-r1),zsδ(-r2),…,zsδ(-rm),u(s))ds+∑0<tk<τ-δT(τ-δ-tk)Ike(tk,zδ(tk),uδ(tk))}+∫τ-δτT(τ-s)Bω(s)uδ(s)ds+∫τ-δτT(τ-s)F(s,zδ(s),zsδ(-r1),zsδ(-r2),…,zsδ(-rm),u(s))ds=T(δ)z(τ-δ)+∫τ-δτT(τ-s)Bω(s)vδ(s))ds+∫τ-δτT(τ-s)F(s,zδ(s),zsδ(-r1),zsδ(-r2),…,zsδ(-rm),vδ(s))ds.

So,

zδ(τ)=T(δ)z(τ-δ)+∫τ-δτT(τ-s)Bω(s)vδ(s)ds+∫τ-δτT(τ-s)F(s,zδ(s),zsδ(-r1),zsδ(-r2),…,zsδ(-rm),vδ(s))ds.

The corresponding solution yδ(t)=y(t,τ-δ,z(τ-δ),vδ) of the initial value problem (16) at time τ, for the control vδ and the initial condition z0=z(τ-δ), is given by:

yδ(τ)=T(δ)z(τ-δ)+∫τ-δτT(τ-s)Bω(s)vδ(s)ds,

and from Lemma 2, we get a solution of the linear initial value problem (10) such that

yδτ-z1<ε2

Therefore,

zδτ-z1 ≤ε2+∫τ-δτTτ-s Fs,zδs,zsδ-r1,zsδ-r2,…,zsδ-rm,vδsds.

Now, since 0<δ<r and τ-δ≤s≤τ, then s-r≤τ-r<τ-δ and

zδ(s-r)=z(s-r).

Hence, there exists δ small enough such that 0<δ<min{r,τ-tp} and

zδτ-z1 ≤ε2+∫τ-δτTτ-s Fs,zs, zs-r1,zs-r2,…,zs-rm,vδsds≤ε2+∫τ-δτTτ-s σ z(s)F12 +∑l=1mzs(-r1)    ds <ε2+ε2= ε

This completes the proof of the Theorem.