Relative controllability of impulsive multi-delay differential systems

In this paper, relative controllability of impulsive multi-delay differential systems in finite dimensional space are studied. By introducing the impulsive multi-delay Gramian matrix, a necessary and sufficient condition, and the Gramian criteria, for the relative controllability of linear systems is given. Using Krasnoselskii’s fixed point theorem, a sufficient condition for controllability of semilinear systems is obtained. Numerically examples are given to illustrate our theoretically results.


Introduction
In many motion processes of nature, science, and technology, the state of motion may be changed or interfered suddenly in a very short time, and then the system state will be changed. If the state change time of the disturbed system is very short, it can be regarded as instantaneous, and then this kind of instantaneous sudden change phenomenon is called pulse phenomenon. Time-delay systems are systems with aftereffect or dead time, genetic systems, equations with deviating arguments or differential difference equations. They are used to model various phenomena from population systems, viscoelasticity, biological sciences, chemistry, economics, mechanics, physics, physiology, and engineering sciences. In the real world, impulsive phenomena and time-delay effects are intertwined and interact with each other. Impulse technology is widely used in the state control of time-delay systems and has applications in military and civil fields.
The delayed exponential matrix functions approach was presented in [6,10] for discrete and continuous delay systems with permutable matrices, respectively. This new approach has been used in the stability of solutions and control problems for linear and nonlinear delay systems (see [1-5, 7-9, 11, 13-20]).
Medved' and Pospíšil extended the idea of deriving the representation of delay differential equations in [6,10] to multi-delay differential equations with linear parts defined by pairwise permutable matrices in [16] and obtained sufficient conditions for the asymptotic stability of solutions. You and Wang [22,23] extended the multiple delayed exponential matrix function in [10] to the impulsive case and used it to discuss the representation and stability of solutions in [24]. However, there are still very few results for the relative controllability of impulsive multi-delay differential systems. In this paper, we study the following impulsive multi-delay differential systems: ν (t) = Aν(t) + n m=1 B m ν(t − ϑ m ) + f t, ν(t) + Cu(t), t ∈ J, t / ∈ T , ν(t) = ψ(t), −ϑ t 0, ϑ := max{ϑ 1 , . . . , ϑ n }, where represent respectively the right and left limits of ν(t) at t = t i .
First, we investigate the relative controllability of the linear case of (1), i.e., f = 0 ∈ R N using the impulsive multi-delayed matrix exponential in (2). Next, we construct a suitable control function for (1), which means that we give a condition (necessary and sufficient) for u ∈ L 2 (J, R N ) to lead the solution of (1) with f = 0 to ν τ1 at the time τ 1 . We apply Krasnoselskii's fixed point theorem to show that (1) is also relatively controllable under suitable conditions. The rest of this paper is organized as follows. In Section 2, we give some notations, concepts, and important lemmas. In Section 3, we establish relative controllability results for linear and semilinear systems, respectively. Examples are given to illustrate our main results in the final section.

Preliminaries
Let R N be the N -dimensional Euclid space with the vector norm · , and R N ×N be the N × N matrix space with real value elements. For ν ∈ R N and A ∈ R N ×N , we introduce the vector infinite-norm ν = max 1 i N |ν i | and the matrix infinite-norm A = max 1 i N N j=1 |a ij |, respectively, where ν i and a ij are the elements of the vector ν and matrix A. Let L(R N ) be the space of bounded linear operators in R N . Denote by C(J, R N ) the Banach space of vector-value bounded continuous functions from J → R N endowed with the norm ν C = sup t∈J ν(t) . In addition, Let X 1 , X 2 be two Banach spaces, and L b (X 1 , X 2 ) denotes the space of all bounded linear operators from X 1 to X 2 . Next, L p (J, X 2 ) denotes the Banach space of functions y : J → X 2 , which are Bochner integrable normed by y L p (J,X2) for some 1 < p < ∞.
For l = 0, by Lemma 1, For l = 1, using Lemma 1, we have For l = k, we suppose that For l = k + 1, using Lemma 1, we have Thus, we obtain (5). Finally, using (3) and (5) via e At e A t , one derives (6) immediately. The proof is finished.

Lemma 3 [Krasnoselskii's fixed point theorem].
(See [12].) Let B be a bounded closed and convex subset of Banach space X, and let F 1 , F 2 be maps of B into X such that F 1 x + F 2 y ∈ B for every pair x, y ∈ B. If F 1 is a contraction and F 2 is compact and continuous, then the equation  (1) is called relatively controllable if for an arbitrary initial vector function ψ ∈ C 1 ([−ϑ, 0], R N ), the final state of the vector ν τ1 ∈ R N and time τ 1 , there exists a control u ∈ L 2 (J, R N ) such that system (1) has a solution ν ∈ C 1 ([−ϑ, 0] ∪ J, R N ) that satisfies the boundary conditions ν and ν(τ 1 ) = ν τ1 .

Linear systems
Let f (t, ν(t)) ≡ 0, t ∈ J. System (1) reduces to the following linear impulsive multidelay controlled system: The solution has a form Similar to the classical Gramian matrix, we consider the impulsive multi-delay Gramian matrix as follows: (7) is relatively controllable if and only if W ϑ1,...,ϑn [0, τ 1 ] is nonsingular.

Semilinear systems
We assume the following: (H1) The operator W : Theorem 3. Suppose that (H1) and (H2) are satisfied. Then system (1) is relatively controllable, provided that Proof. Using hypothesis (H1), for arbitrary ν(·) ∈ PC and t ∈ J, we define the control function u ν (t) by We show that, using this control, the operator F : PC → PC , defined by has a fixed point ν, which is a mild solution of (1).
For each positive number r, let B r = {ν ∈ PC : ν PC r} (a bounded, closed, and convex set of PC ). Set R f = sup t∈J f (t, 0) .
We divide the proof into three steps.
Step 1. We claim that there exists a positive number r such that F(B r ) ⊆ B r . From (H2) and Hölder's inequality we obtain that From (12), (H1) and (H2) we have
Hence, we obtain F(B r ) ⊆ B r for such an r. Now, we define operators F 1 and F 2 on B r as Step 2. We claim that F 1 is a contraction mapping. Let ν, γ ∈ B r . From (H1) and (H2), for each t ∈ J, we have Thus, so we obtain From (11) we have T < 1, so F 1 is a contraction.
Step 3. We claim that F 2 : B r → PC is a compact and continuous operator. Let ν n ∈ B r with ν n → ν in B r . Using (H2), we have f (s, ν n (s)) → f (s, ν(s)) in PC , and thus, using the Lebesgue dominated convergence theorem, we have To check the compactness of F 2 : B r → PC , we prove that F 2 (B r ) is equicontinuous and uniformly bounded. In fact, for any ν ∈ B r , t k < t t + d t k+1 , k = 0, 1, . . . , h, where I is the identity matrix.
From above we see that Now, we only need to check S i (t) → 0 as d → 0, i = 1, 2, 3. Clearly, As a result, we immediately obtain that for all ν ∈ B r . Therefore, F 2 (B r ) is equicontinuous in PC .
https://www.journals.vu.lt/nonlinear-analysis Next, repeating the above computations, we have Hence, F 2 (B r ) is uniformly bounded. From Theorem 1, F 2 (B r ) is relatively compact in PC . Thus, F 2 : B r → PC is a compact and continuous operator.

Conclusion
In this paper the relative controllability of impulsive multi-delay differential systems in finite-dimensional space is considered. In [24] the authors construct the index of impulsive multi delay matrix and give the explicit solution of linear impulsive multi delay differential equations. Based on the expression of the solution of linear impulsive multi delay differential equations, necessary and sufficient conditions for the relative controllability of linear systems and the Gramian criteria are given. In Theorem 3, using Krasnoselskii fixed point theorem, we give a sufficient condition for the controllability of semilinear systems.
In Theorem 2 the control function is given, but it is not necessarily optimal, and we hope in the future to study the optimal control problem of impulsive multi-delay differential equations. In Theorem 3, we require the operator F 2 to be compact, and we hope to study controllability under noncompact conditions in the future.