Relative controllability of impulsive multi-delay differential systems*

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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 differ- ential 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:

[1]

where are constant matrices, and for each and Now and the control function takes values from represent respectively the right and left limits of

First, we investigate the relative controllability of the linear case of (1), i.e., 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 to lead the solution of (1) with at the time. 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.

Let be the -dimensional Euclid space with the vector norm, and be the matrix space with real value elements. For and, we introduce the vector infinite-norm and the matrix infinite-norm respectively, where and are the elements of the vector and matrix. Let be the space of bounded linear operators in . Denote by the Banach space of vector-value bounded continuous functions from endowed with the norm In addition, We introduce a spaceDenote there exist and with for any and Let be two Banach spaces, and denotes the space of all bounded linear operators from Next, denotes the Banach space of functions which are Bochner integrable normed by for some

We recall the notation of the multi-delayed matrix exponential given by [16]:

[2]

where and is the zero matrix.

From [24] we knowand

(3)

Where

Next, the solution of (1) has the form

(4)

Lemma 1. (See [16, Lemma 13].) If then

Lemma 2.Suppose that is convergent, For any we have

(5)

(6)

Proof. Without loss of generality, we suppose that and We use mathematical induction.

For, by Lemma 1,

For, using Lemma 1, we have

For we suppose that

For using Lemma 1, we have

Thus, we obtain (5).

Finally, using (3) and (5) via one derives (6) immediately. The proof is finished.

Lemma 3 [Krasnoselskii’s fixed point theorem]. (See [12].) Let be a bounded closed and convex subset of Banach space, and let . be maps of into such that for every pair If F₁is a contraction and F₂is compact and continuous, then the equationhas a solution on .

Theorem 1 [PC -type Ascoli–Arzela theorem]. (See [21, Thm. 2.1].) Let where is a Banach space. Then is a relatively compact subset of

1. (i) is a uniformly bounded subset of

2. (ii) is equicontinuous in

3. (iii) are relatively compact subsets of X.

Definition 1. (See [11, Def. 4].) System (1) is called relatively controllable if for an arbitrary initial vector function the final state of the vector and time, there exists a control such that system (1) has a solution that satisfies the boundary conditions and

3.1 Linear systems

Let System (1) reduces to the following linear impulsive multi- delay controlled system:

[7]

The solution has a form

Similar to the classical Gramian matrix, we consider the impulsive multi-delay Gramian matrix as follows:

Theorem 2. System (7) is relatively controllable if and only if is nonsingular.

Proof. First, we verify the sufficiency. Since is nonsingular, its inverse is well defined. For any final state one can select a control function as follows:

Where

Then

Next, by contradiction we prove the necessity. We assume that is singular matrix, i.e., there exists at least one nonzero state such that

Then one obtains

which implies for all

Since system (7) is relatively controllable, according to Definition 1, there exists a control that drives the initial state to zero at , i.e.,

[8]

Similarly, there also exists a control that drives the initial state to (nonzero) at , i.e.,

[9]

Then from (8) and (9) we have

[10]

Multiplying both sides of (10) by , we obtain

Thus, which conflicts with Thus, the impulsive multi-delay Gramian matrix is nonsingular. The proof is complete.

3.2 Semilinear systems

We assume the following:

(H1)

(H2)

Theorem 3. Suppose that (H1) and (H2) are satisfied. Then system (1) is relatively controllable, provided that

[11]

where

Proof. Using hypothesis (H1), for arbitrary and we define the control function by

[12]

We show that, using this control, the operator, defined by

has a fixed point , which is a mild solution of (1).

We check that which means thatsteers system (1) from to in infinite time This implies that system (1) is relative vontrollable on J.

For each positive number (a bounded, closed, and convex set of PC ). Set .

We divide the proof into three steps.

Step 1. We claim that there exists a positive number such that From (H2) and Hölder’s inequality we obtain that

From (12), (H1) and (H2) we have

Where

From (H1) and (H2) we have

Where

Hence, we obtain for such an .

Now, we define operators F₁ and F₂ on as

And

Step 2. We claim that F1 is a contraction mapping.

Let From (H1) and (H2), for each we have

Thus,

so we obtain

From (11) we have T < 1, so F₁ is a contraction.

Step 3. We claim that is a compact and continuous operator.

Let with. Using (H2), we have in PC , and thus, using the Lebesgue dominated convergence theorem, we have

which implies that F₂ is continuous on .

To check the compactness of, we prove that is equicontinuous and uniformly bounded. In fact, for any

Let

where is the identity matrix.

From above we see that

Now, we only need to check Clearly,

By the continuity of we have Also,

As a result, we immediately obtain that

for all Therefore,is equicontinuous in PC .

Next, repeating the above computations, we have

Hence, is uniformly bounded. From Theorem 1, is relatively compact in PC . Thus,is a compact and continuous operator.

Furthermore, using Theorem 3, has a fixed point on . Obviously, is a solution of system (1) satisfying. The boundary condition holds from (4). The proof is complete.

Example 1. Consider the following semilinear impulsive multi-delay differential controlled system:

[13]

and set Then and

Noteare mutually permutable for and

where

Specifically,

Then

Further, for any ν, µ ∈ R2,

Note

Thus all the conditions of Theorem 3 are satisfied, so (13) is relatively controllable on [0,0.6]; see Fig. 1.

Figure 1
The state trajectories of ν(t) in [0, 1] when u = [0.8t, 0.9t]T in Example 1.

Example 2. In Example 1, let Note is a nonsingular matrix. From Theorem 2 we know that the linear multi-delay system is

and then the control function is given by

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 to be compact, and we hope to study controllability under noncompact conditions in the future.