Linear impulsive dynamic systems on time scales

The purpose of this paper is to present the fundamental con- cepts of the basic theory for linear impulsive systems on time scales. First, we introduce the transition matrix for linear impulsive dynamic systems on time scales and we establish some properties of them. Second, we prove the existence and uniqueness of solutions for linear impulsive dynamic systems on time scales. Also we give some sufficient conditions for the stability of linear impulsive dy- namic systems on time scales.


Introduction
Differential equations with impulse provide an adequate mathematical description of various real-word phenomena in physics, engineering, biology, economics, neutral network, social sciences, etc. Also, the theory of impulsive differential equations is much richer than the corresponding theory of differential equations without impulse effects. In the last fifty years the theory of impulsive differential equations has been studied by many authors. We refer to the monographs [10]- [12], [27], [33], [42] and the references therein.
In this paper we study some aspects of the qualitative theory of linear impulsive dynamic systems on time scales. In Section 2 we present some preliminary results on linear dynamic systems on time scales and also we give an impulsive inequality on time scales. In Section 3 we prove the existence and uniqueness of solutions for homogeneous linear impulsive dynamic systems on time scales. For this, we introduce the transition matrix for linear impulsive dynamic systems on time scales and we give some properties of the transition matrix. In Section 4 we prove the existence and uniqueness of solutions for nonhomogeneous linear impulsive dynamic systems on time scales. In Section 5 we give some sufficient conditions for the stability of linear impulsive dynamic systems on time scales. Finally, in Section 6 we present a brief summary of time scales analysis.

Preliminaries
Let R n be the space of n-dimensional column vectors x = col(x 1 , x 2 , ...x n ) with a norm ||·||. Also, by the same symbol ||·|| we will denote the corresponding matrix norm in the space M n (R) of n × n matrices. If A ∈ M n (R), then we denote by A T its conjugate transpose. We recall that ||A|| := sup{||Ax||; ||x|| ≤ 1} and the following inequality ||Ax|| ≤ ||A|| · ||x|| holds for all A ∈ M n (R) and x ∈ R n . A time scales T is a nonempty closed subset of R. The set of all rd-continuous functions f : T → R n will be denoted by C rd (T, R n ).
The notations [a, b], [a, b), and so on, will denote time scales intervals such as [a, b] := {t ∈ T; a ≤ t ≤ b}, where a, b ∈ T. Also, for any τ ∈ T, let T (τ ) := [τ, ∞) ∩ T and T + := T (0) . Then We denote by R (respectively R + ) the set of all regressive (respectively positively regressive) functions from T to R.
The space of all rd-continuous and regressive functions from T to R is denoted by C rd R(T, R). Also, C + rd R(T, R) := {p ∈ C rd R(T, R); 1 + µ(t)p(t) > 0 for all t ∈ T}.
We denote by C 1 rd (T, R n ) the set of all functions f : T → R n that are differentiable on T and its delta-derivative f ∆ (t) ∈ C rd (T, R n ). The set of rdcontinuous (respectively rd-continuous and regressive) functions A : T → M n (R) is denoted by C rd (T, M n (R)) (respectively by C rd R(T, M n (R))). We recall that a matrix-valued function A is said to be regressive if I + µ(t)A(t) is invertible for all t ∈ T, where I is the n × n identity matrix. Now consider the following dynamic system on time scales where A ∈ C rd R(T + , M n (R)). This is a homogeneous linear dynamic system on time scales that is nonautonomous, or time-variant. The corresponding nonhomogeneous linear dynamic system is given by where h ∈ C rd (T + , R n ).
A function x ∈ C 1 rd (T + , R n ) is said to be a solution of (2.2) on T + provided If A ∈ C rd R(T + , M n (R)) and h ∈ C rd (T + , R n ), then for each (τ, η) ∈ T + × R n the initial value problem A matrix X A ∈ C rd R(T + , M n (R)) is said to be a matrix solution of (2.1) if each column of X A satisfies (2.1). A fundamental matrix of (2.1) is a matrix solution X A of (2.1) such that det X A (t) = 0 for all t ∈ T + . A transition matrix of (2.1) at initial time τ ∈ T + is a fundamental matrix such that X A (τ ) = I. The transition matrix of (2.1) at initial time τ ∈ T + will be denoted by Φ A (t, τ ). Therefore, the transition matrix of (2.1) at initial time τ ∈ T + is the unique solution of the following matrix initial value problem [15,Theorem 5.24]) If A ∈ C rd R(T + , M n (R)) and h ∈ C rd (T + , R n ), then for each (τ, η) ∈ T + × R n the initial value problem has a unique solution x : T (τ ) → R n given by EJQTDE, 2010 No. 11, p. 3 As in the scalar case, along with (2.1), we consider its adjoint equation If A ∈ C rd R(T + , M n (R)), then the initial value problem y ∆ = −A T (t)x σ , x(τ ) = η, has a unique solution y : T (τ ) → R n given by and h ∈ C rd (T + , R n ), then for each (τ, η) ∈ T + × R n the initial value problem has a unique solution x : T (τ ) → R n given by

An application of Lemma
which implies (2.5).

Homogeneous linear impulsive dynamic system on time scales
Consider the following homogeneous linear impulsive dynamic system on time scales We assume for the remainder of the paper that, for k = 1, 2, ..., the points of impulse t k are rigth-dense.
Along with (3.1) we consider the following initial value problem We note that, instead of the usual initial condition x(τ ) = η, we impose the limiting condition x(τ + ) = η which, in general case, is natural for (3.2) since (τ, η) may be such that τ = t k for some k = 1, 2, .... In the case when τ = t k for any k, we shall understand the initial condition x(τ + ) = η in the usual sense, that is, x(τ ) = η.
Theorem 3.1. If A ∈ C rd R(T + , M n (R)) and B k ∈ M n (R), k = 1, 2, ..., then any solution of (3.2) is also a solution of the impulsive integral equation and conversely.
Proof. There exists i ∈ {1, 2, ...} such that τ ∈ [t i−1 , t i ). Then any solution of (3.2) on [τ, t i ) is also a solution of integral equation Further, any solution of the initial value problem is a solution of integral equation . It follows that Next, we suppose that, for any k > i + 2, any solution of (3.2) on [t k−1 , t k ) is a solution of (3.3). Then any solution of the initial value problem is a solution of integral equation . It follows that Therefore, by the Mathematical Induction Principle, (3.3) is proved. The converse statement follows trivially and the proof is complete.
Theorem 3.2. If A ∈ C rd R(T + , M n (R)) and B k ∈ M n (R), k = 1, 2, ..., then the solution of (3.2) satisfies the following estimate Then, by Lemma 2.1, it follows that Since for any a ≥ 0, lim uցµ(s) then explicit estimation of the modulus of the exponential function on time scales (see [24]) gives Thus we obtain (3.4).
is the transition matrix of (2.1) at initial time s ∈ T + .
In the following, we will assume that I + B k is invertible for each k = 1, 2, ....
.., then for each (τ, η) ∈ T + × R n the initial value problem (3.2) has a unique solution given by Further, we consider the initial value problem . This initial value problem has the unique solution given by ). Next, we suppose that, for any k > i + 2, the unique solution of (3.2) on [t k−1 , t k ] is given by Then the initial value problem . It follows that and so . Therefore, by the Mathematical Induction Principle, the theorem is proved.
Moreover, the following properties hold: From the Theorems 3.2 and 3.3, we obtain the following result.
.., then we have the following estimate Proof. Using the Theorems 3.2 and 3.3, for all τ, t ∈ T + with t ∈ T (τ ) , it follows Let X A (t), t ∈ T + , be the unique solution of (3.6) with the initial condition Theorem 3.4. If A ∈ C rd R(T + , M n (R)) and B k ∈ M n (R), then the impulsive transition matrix S A (t, s) has the following properties EJQTDE, 2010 No. 11, p. 9 for each t k ≥ s. Therefore, Y (t) = X A (t)X −1 A (s) solves the initial value problem (3.6), which has exactly one solution. Therefore, The properties (ii) and (iii) follows from (i). Now, from Theorem A.2 and Corollary 3.1, we have that Proof. Indeed, from Theorem 3.4 and Theorem A.2, we have According to Theorem A.2 and (6.5), we have which has exactly one solution. It follows that S ⊖A T (t, s) = Y (t) = (S −1 A (t, s)) T , and so S −1 Indeed, we have Since A is regressive, then A T is also regressive. Then, by (6.5), the above equality is equivalent to ... The homogeneous linear impulsive dynamic system on time scales (3.9) is called the adjoint dynamic system of (3.2). Corollary 3.3. If A ∈ C rd R(T + , M n (R)) and B k ∈ M n (R), k = 1, 2, ..., then any solution of (3.9) is also a solution of the impulsive integral equation and conversely.
and h ∈ C rd R(T + , R n ) then, for each (τ, η) ∈ T + × R n , the initial value problem (4.1) has a unique solution given by Then the unique solution of (4.1) on [τ, t i ] is given by For t ∈ (t i , t i+1 ] the Cauchy problem has the unique solution It follows that and so Next, we suppose that, for any k > i + 2, the unique solution of (4.1) on [t k−1 , t k ] is given by Then the initial value problem has the unique solution Using the Remark 3.1, we obtain that and so, Therefore, by the Mathematical Induction Principle, (4.2) is proved.
Corollary 4.1. If B k ∈ M n (R), k = 1, 2, ..., A ∈ C rd R(T + , M n (R)), and h ∈ C rd R(T + , R n ) then, for each (τ, η) ∈ T + × R n , the initial value problem has a unique solution given by EJQTDE, 2010 No. 11, p. 14 Theorem 4.2. If B k ∈ M n (R), k = 1, 2, ..., A ∈ C rd R(T + , M n (R)), c := {c k } ∞ k=1 ∈ l ∞ (R n ) and h ∈ C rd R(T + , R n ) then, for each (τ, η) ∈ T + × R n , the initial value problem has a unique solution given by Since A is regressive, then A T is also regressive, and thus the above inequality is equivalent to

Boundedness and stability of linear impulsive dynamic system on time scales
Definition 5.1. The dynamic system (3.2) is said to be exponentially stable (e.s.) if there exists a positive constant λ with −λ ∈ R + such that for every τ ∈ T + , there exists N = N (τ ) ≥ 1 such that the solution of (3.2) through (τ, x(τ )) satisfies The dynamic system (3.2) is said to be uniformly exponentially stable (u.e.s.) if it is e.s. and the constant N can be chosen independently of τ ∈ T + . Theorem 5.1. Suppose that B k ∈ M n (R), k = 1, 2, ..., A ∈ C rd R(T + , M n (R)), and there exists a positive constant θ such that t k+1 − t k < θ, k = 1, 2, ....

If the solution of initial value problem
Proof. From Theorem 4.1, the solution of (5.1) is given by For each fixed t ∈ T (τ ) , by the Corollary 3.2, the operator U t : l ∞ (R n ) → R n , given by U t (c) := x(t), is a bounded linear operator. In fact, ||U t (c)|| ≤ τ <tj <t ||S A (t, t + j )|| · ||c|| l ∞ < ∞ for any c ∈ l ∞ (R n ). Since the solution x(t) of (5.1) is bounded for any c ∈ l ∞ (R n ), then uniform boundedness principle implies that there exists a constant K > 0 such that ||x(t)|| ≤ K||c|| l ∞ for all c ∈ l ∞ (R n ) and τ ∈ T + , that is, Let τ ∈ T + be fixed. Then there exists i ∈ {1, 2, ...} such that τ ∈ [t i−1 , t i ). We define the sequences {β j } ∞ j=1 and {c j } ∞ j=1 ∈ l ∞ (R n ) given by Here y is an arbitrary fixed element of R n \{0}. Also, we observe that ||c|| l ∞ = 1. Hence, from (5.3) we obtain Since y is an arbitrary element of R n \{0} then it follows that Now, if t k > t i and t ∈ [τ, t k+1 ) then, from (5.4), we obtain that Without loss of generality we can assume that K > 1.
Now we have to show that N k can be chosen independently of k. From Corollary 3.2 and (5.7), we obtain that ||S ] then, by Lemma 5.1, we can choose a constant λ with 1 λ > θ such that that −λ ∈ R + . Using the Bernoulli's inequality, we have that and thus 1−λθ e M }. Next, let τ ∈ T + be arbitrary. Then there exists k ∈ {1, 2, ...} such that τ ∈ [t k , t k+1 ). Then we have Therefore ||S A (t, τ )|| ≤ N e −λ (t, τ ), with N = N 2 , and the theorem is proved.

If the solution of initial value problem (5.1) is bounded for any
, then the solution of (4.1) is bounded for each h ∈ BC rd R(T + , R n ).
Proof. Let i ∈ {1, 2, ...} be such that τ ∈ [t i , t i+1 ) By Theorem 5.2 there exist positive constants N and λ with −λ ∈ R + such that For every function h ∈ C rd R(T + , R n ), the corresponding solution x h of (4.1) is given by (4.2). From (4.2) we obtain that We have that t τ N ||h(s)||e −λ (t, σ(s))∆s Further, let t k be the greatest of all t j < t. Then and, by Lemma 5.2, we have that Therefore, Since lim t→∞ e −λ (t, τ ) = 0 it follows that x h is bounded.

It follows that
where N = max{a, b}.
(i) f is continuous at every right-dense point t ∈ T, (ii) f (t − ) := lim s→t − f (s) exists and is finite at every left-dense point t ∈ T.
The set of all rd-continuous functions f : T →R n will be denoted by C rd (T,R n ). A function p : T →R is said to be regressive (respectively positively regressive) if 1 + µ(t)p(t) = 0 (respectively 1 + µ(t)p(t) > 0) for all t ∈ T. The set R (respectively R + ) of all regressive (respectively positively regressive) functions from T to R is an Abelian group with respect to the circle addition operation ⊕, given by (p ⊕ q)(t) := p(t) + q(t) + µ(t)p(t)q(t).
The inverse element of p ∈ R is given by and so, the circle subtraction operation ⊖ is defined by The space of all rd-continuous and regressive functions from T to R is denoted by C rd R(T,R). Also, The set of rd-continuous (respectively rd-continuous and regressive) functions A : T → M n (R) is denoted by C rd (T, M n (R)) (respectively by C rd R(T, M n (R))). We recall that a matrix-valued function A is said to be regressive if I + µ(t)A(t) is invertible for all t ∈ T, where I is the n × n identity matrix. Moreover, the set R(T, M n (R)) of all regressive matrix-valued functions is a group with respect to the addition operation ⊕ define for all t ∈ T. The inverse element of A ∈ R(T, M n (R)) is given by for all t ∈ T.
Definition A.2. A function f : T →R n is said to be differentiable at t ∈ T, with delta-derivative f ∆ (t) ∈R n if given ε > 0 there exists a neighborhood U of t such that, for all s ∈ T, where f σ (t) := f (σ(t)) for all t ∈ T.
We denote by C 1 rd (T,R n ) the set of all functions f : T →R n that are differentiable on T and its delta-derivative f ∆ (t) ∈ C rd (T,R n ).
(i) If f is differentiable at t, then f is continuous at t.
(ii) If f is continuous at t and t is right-scattered, then f is differentiable at t with (iii) If f is differentiable at t and t is right-dense, then (iv) If f is differentiable at t, then f σ (t) = f (t) + µ(t)f ∆ (t).
(v) If f, g : T →R n are both differentiable at t, then the product f T g is also differentiable at t and ( Definition A.3. Let f ∈ C rd (T,R n ). A function g : T → R n is called the antiderivative of f on T if it is differentiable on T and satisfies g ∆ (t) = f (t) for all t ∈ T. In this cases, we define t a f (s)∆s = g(t) − g(a), a, t ∈ T. ( Let p ∈ C rd R(T,R) and µ(t) = 0 for all t ∈ T. Then the exponential function on T is defined by e p (t, s) = exp and it is the unique solution of the initial value problem y ∆ = p(t)y, y(s) = 1.