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Approximate Controllability of Semilinear Impulsive Evolution Equations
We prove the approximate controllability of the following semilinear impulsive evolution equation: where , and are Hilbert spaces, , is a bounded linear operator, are smooth functions, and is an unbounded linear operator in which generates a strongly continuous semigroup . We suppose that is bounded and the linear system is approximately controllable on for all . Under these conditions, we prove the following statement: the semilinear impulsive evolution equation is approximately controllable on .
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: where , and are Hilbert spaces, , is a bounded linear operator, are smooth functions, and is an unbounded linear operator in which generates a strongly continuous semigroup .
Definition 1 (approximate controllability). System (1) is said to be approximately controllable on if for every , there exists such that the solution of (1) corresponding to verifies (see Figures 1 and 2)
We assume the following main hypotheses:(A) is a bounded function;(B)linear system without impulses (8) is approximately controllable on for all .
This paper has been motivated by the works done in Bashirov and Ghahramanlou , Bashirov and Jneid , and Bashirov et al. , where a new technique to prove the controllability of evolution equations without impulses is used avoiding fixed point theorems, and the work done in .
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  studied the exact controllability of impulsive partial neutral functional differential equations with infinite delay, Radhakrishnan and Balachandran  studied the exact controllability of semilinear impulsive integrodifferential evolution systems with nonlocal conditions, and Selvi and Mallika Arjunan  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  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 does not depend on the control variable, and Leiva and Merentes in  studied the approximate controllability of the semilinear impulsive heat equation using the fact that the semigroup generated by is compact.
In this paper, we are not assuming the compactness of the semigroup generated by the unbounded operator ; when this semigroup is compact we can consider weaker condition on the nonlinear perturbation and in the linear part of the system without impulses. Specifically, we can assume the following hypotheses:(a) with , , ;(b)the linear system is approximately controllable only on .
This case is similar to the semilinear impulsive heat equations studied in , where the authors use conditions (a) and (b), the compactness of the semigroup generated by the Laplacian operator , and Rothe’s fixed point theorem to prove the approximate controllability of the system on .
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 portion of the domain for the spatial variable, then the system is approximately controllable only on for , which was proved in , where the following system governed by the wave equations was studied: where is a bounded domain in , is an open nonempty subset of , denotes the characteristic function of the set , the distributed control , and , .
However, if the control acts on the whole domain , it was proved in  that the system is controllable , for all . More specifically, the authors studied the following system: where is a bounded domain in , the distributed control , and , .
To justify the use of this new technique , in this paper, we consider as an application the semilinear impulsive wave equation with controls acting on the whole domain , so that hypotheses (A) and (B) hold: where , is a bounded domain in , the distributed control , , , and are smooth functions with being bounded.
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 all and the initial value problem admits only one mild solution given by
Definition 2. For system (8), one defines the following concept: the controllability maps , , defined by satisfy the following relation: The adjoints of these operators , are given by The Gramian controllability operators are given by (3) and
Lemma 3. The following statements are equivalent to the approximate controllability of the linear system (8) on . (a).(b).(c), in .(d).(e)For all , one has , where So and the error of this approximation is given by the formula (f)Moreover, if one considers for each the sequence of controls given by one gets that with the error of this approximation given by the formula
Remark 4. The foregoing lemma implies that the family of linear operators , defined for by is an approximate inverse for the right of the operator , in the sense that in the strong topology.
Lemma 5. if and only if linear system (8) is controllable on . Moreover, given an initial state and a final state , one can find a sequence of controls : such that the solutions of the initial value problem satisfy that is,
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 and , the initial value problem admits only one mild solution given byNow, 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 .
Proof. Given an initial state , a final state , and , we want to find a control steering the system from to an -neighborhood of on time . Specifically, Consider any and the corresponding solution of initial value problem (25). For , we define the control as follows: where Now, assume that . Then the corresponding solution of initial value problem (25) at time can be written as follows: So, The corresponding solution of initial value problem (22) at time is given by Therefore, Then, putting and , we obtain the following estimate: Hence, From Lemma 5, there exists such that So, taking , we obtain for the corresponding control that Geometrically, the proof goes as shown in Figure 3.
This completes the proof of the theorem.
As an application, we will prove the approximate controllability of the following control system governed by the semilinear impulsive wave equation: where is a bounded domain in , the distributed control , , , and are smooth functions with being bounded.
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. Let and consider the linear unbounded operator defined by , where Then the eigenvalues of have finite multiplicity equal to the dimension of the corresponding eigenspace and . Moreover, consider the following. (a) There exists a complete orthonormal set of eigenvectors of . (b) For all , we have where is the usual inner product in and which means the set is a complete family of orthogonal projections in and , . (c) generates an analytic semigroup given by (d) The fractional powered spaces are given by with the norm Also, for , we define , which is a Hilbert space endowed with the norm given by
Then, (1) can be written as abstract second order ordinary differential equations in as follows: where are defined by With the change of variables , we can write second order equation (46) as a first order system of ordinary differential equations in the Hilbert space as follows: where , , is an unbounded linear operator with domain , and are defined by
Proposition 7. The group generated by the operator has the following representation: where is a complete family of orthogonal projections in the Hilbert space given by
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 with bounded nonlinear perturbation.
Theorem 8. Semilinear impulsive wave equation (38) is approximately controllable on .
Proof. From , we know that the corresponding linear system without impulses is exactly controllable on , for all . On the other hand, since is bounded, there exists such that . Therefore, we obtainfor all with the Lebesgue measure of . 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 for and the nonlinear perturbation is bounded. Example of such systems is the following controlled thermoelastic plate equation whose linear part was studied in : in the space , where is a bounded domain in , the distributed controls , and are smooth functions with , , being bounded.
Conflict of Interests
The author declares that there is no conflict of interests regarding the publication of this paper.
This work has been supported by CDCHT-ULA-C-1796-12-05-AA and BCV.
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