Roughness of the Controllability for Time Varying Systems Under the Influence of Impulses , Delay , and Nonlocal Conditions

In this paper, we prove that the controllability of time varying linear system is preserved if we add impulses, delay and nonlocal conditions on it. In order to do that, we assume some conditions on the non-linear terms and apply the Rothe’s fixed point theorem to obtain the main result. In other words, we find sufficient controllability conditions for the semilinear system under the influence of impulses, delay, and nonlocal conditions.

where J = [ , τ] and J ′ = J\{t , t , . . . , tp}, endowed with the norm The following notation will be used: Analogously, we de ne the space PWrq([−r, ]; (R n ) q ) endowed with the norm On the other hand, the following Banach space is considered as well In R n × R m the following norm is considered: Given an arbitrary (z, u) , the following quantity is also considered: Also we consider the norm of B as follows In addition, in concordance with the semilinear system (1.1), the following linear system is considered: Now, we give the most important de nition of this work, which itself is the main objective of it, to study the controllability of the semilinear system (1.1): There are a lot of works about the controllability of the linear (1.6), perhaps one can see [4], [18] and [34]. Nevertheless, the reference for semilinear systems is not extensive, in this regard, we can refer to the papers done by Lukes in [27], J.C. Coron in [10](see Theorem 3.40 and Corollary 3.41 ). On the other hand, Vidyasager in [35] proved the controllability by using Schauder Fixed Point Theorem and f does not depend u ∈ R m . However, Dauer in [13] found conditions on f to prove the controllability of the semilinear system (without impulses and nonlocal conditions). But, V. N. Do [14] weaker the conditions on f and prove the controllability of the system (1.1) ( without impulses, delay and non local conditions) containing Dauer's conditions; it is good to recall that all these conditions depends strongly on the linear system (1.6); especially, on the fundamental matrix Φ(t) of the linear system (2.9), which is in general not available in closed form.
There are others concepts of controllability, perhaps, the local controllability, which has been studied by [4], [5], [6], [7], [8], [9], [28], [29], and [31], but as far as we know, without impulses and non local conditions. The controllability of di erential equation with impulses is in e ervescence now; for evolution equations there are papers by [30] by J.J. Nieto and C.C. Tiesdell, and in [15] by Zhi-Qing Zhu and Qing-Wen Lin. Moreover, in [22] and [23], the Rothe's xed point theorem, which has been applied to prove the controllability of semilinear systems with impulses, is the essential motivation for doing this work. S. Selvi and M. Malika Arjunan in [32] studied the controllability of impulsive di erential systems with nite delay by using measures of noncompactness and Monch's Fixed Point Theorem. For in nite dimensional Banach spaces, we are sure that some ideas presented here can be used to address also the controllability of evolution equations with impulses, delay and nonlocal conditions simultaneously, and the nonlinear term involving all the variables, the time, the state and the control. On the other hand, some results from [3], [21], [24] and [25], can be used.

) Let E be a Banach space. Let B ⊂ E be a closed convex subset such that the zero of E is contained in the interior of B. Let Ψ : B → E be a continuous mapping with Ψ(B) relatively compact in E and Ψ(∂B) ⊂ B. Then there is a point x
Our main hypotheses will be: The controllability of the linear system (1.6), the continuity of the fundamental matrix of the uncontrolled linear system and the conditions (1.2)-(1.5) satis ed by the nonlinear terms f , g, I k .

Controllability of Linear Systems
In this section, we shall present some known characterization for the controllability of linear systems (1.6) without impulses, delay and nonlocal conditions. To this end, we note that for all z ∈ R n and u ∈ admits only one solution given by i.e., the matrix Φ(t) satis es: where I R n is the n × n identity matrix. Therefore, there exist constants M ≥ and ω ≥ such that De nition 2.1. For the system (1.6) we de ne the following concept: The controllability maps (for τ > ) G : The adjoint operators G * : R n −→ L ([ , τ]; R m ) of the operator G is given by (2.13) and the Controllability Gramian operator W : R n → R n is given by Also, we shall use the following result from [11],pp 55, and [12].
Lemma 2.1. Let Y and Z be Hilbert space, G ∈ L(Y , Z) and G * ∈ L(Z, Y) the adjoint operator. Then the following statements holds, (see [17]) Then the following statements are equivalent a) Therefore, the operators Υ : is called the steering operator and it is a right inverse of G, in the sense that and a control steering the system (1.6) from initial state z to a nal state z at time τ > is given by where G| S is the restriction of G to S.

Main Results
In this section, we shall prove the controllability of the nonlinear system (1.1) with impulses, delay, and nonlocal conditions. To this end, for all ϕ ∈ PWr([−r, ]; R n ) and u ∈ C([ , τ]; R m ) according to [19] and [2] the initial value problem admits only one solution given by Now, we de ne the operator

U(t, s)B(s)(ΥL(z, u))(s)ds
and (3.29) The following proposition follows trivially from the de nition of the operator S.  C([ , τ]; R m ) such that for a given ϕ ∈ PWr([−r, ]; R n ), z ∈ R n the corresponding solution z(·) of (1.1) satis es: Proof We shall prove this theorem by claims.

Claim 1.
The operator S is continuous. In fact, to prove the continuity of S, it is enough to prove the continuity of the operators S and S de ned above.
The continuity of S follows from the continuity of the nonlinear functions f (t, zs , u), I k (z, u), g(z) and the following estimate , v(t k )) , where, and, The continuity of the operator S follows from the continuity of the operators L and Υ de ne above.

Claim 2. S maps bounded sets of
Consider the following equality and Since U(t, s) is continuous, U(t , s) − U(t , s) goes to zero as t → t , and so does the sum and the integral from t to t , which implies that S (D) is equicontinuous. Moreover, the equicontinuity of S (D) follows from the continuity of the evolution operator U(t, s). Hence, S maps bounded sets into equicontinuous sets. where Therefore, where M is given by: Hence, Claim 5. The operator S has a xed point. In fact, by Claim 4, we know that for a xed < ρ < there exists R > big enough such that Hence, applying the Proposition 3.1, we get that the nonlinear system (1.1) is controllable on [ , τ]. Moreover, u = ΥL(z, u) = B * (·)U * (τ, ·)W − L(z, u), such that for a given ϕ ∈ C([−r, ]; R n ), z ∈ R n the corresponding solution z(t) = z(t, u) of (1.1) satis es: It nishes the proof. Now, we present another version of the previous theorem, which follows from the estimates considered in Claim 4. Hence, there exists R > such that |S(z, u) | ≤ ρ |(z, u) |, |(z, u) | = R.
Then, analogously to the previous theorem the proof of Theorem 3.2 immediately follows by applying Proposition 3.1.

Final Remark
Some of the ideas presented in this work can be used to prove that under certain conditions the controllability of many control systems is not destroyed if the system is in uenced by impulses, nonlocal conditions and delay simultaneously. In other words, we proved that the controllability is robust in the presence of impulses, delay, and nonlocal conditions. This happens for many real life control systems where impulses, delay, and nonlocal conditions are intrinsic phenomena of the system. Moreover, in several papers we have shown that the in uence of impulses do not destroy the controllability of some known systems like the heat equation, the wave equation and the strongly damped wave equation( [3,[23][24][25]).