EXPONENTIAL STABILITY OF DYNAMIC EQUATIONS ON TIME SCALES

We investigate the exponential stability of the zero solution to a system of dynamic equations on time scales. We do this by deﬁning appropriate Lyapunov-type functions and then formulate certain inequalities on these functions. Several examples are given.


Introduction
This paper considers the exponential stability of the zero solution of the first-order vector dynamic equation (1.1) Throughout the paper, we let x(t,t 0 ,x 0 ) denote a solution of the initial value problem (IVP) (1.1), (1.2) (For the existence, uniqueness, and extendability of solutions of IVPs for (1.1)-(1.2), see [2,Chapter 8].) Also we assume that f : [0,∞) × R n → R n is a continuous function and t is from a so-called "time scale" T (which is a nonempty closed subset of R). Throughout the paper, we assume that 0 ∈ T (for convenience) and that f (t,0) = 0, for all t in the time scale interval [0, ∞) := {t ∈ T : 0 ≤ t < ∞}, and call the zero function the trivial solution of (1.1). If T = R, then x ∆ = x and (1.1)-(1.2) becomes the following IVP for ordinary differential equations 3) x t 0 = x 0 , t 0 ≥ 0. (1.4) we use Lyapunov-type functions on time scales and then formulate appropriate inequalities on these functions that guarantee that the trivial solution to (1.1) is exponentially or uniformly exponentially stable on [0, ∞). Some of our results are new even for the special cases T = R and T = Z. To understand the notation used above and the idea of time scales, some preliminary definitions are needed. Definition 1.1. A time scale T is a nonempty closed subset of the real numbers R.
Since we are interested in the asymptotic behavior of solutions near ∞, we assume that T is unbounded above.
Since a time scale may or may not be connected, the concept of the jump operator is useful. (1.5) and define the graininess function µ : Also let x σ (t) = x(σ(t)), that is, x σ is the composite function x • σ. The jump operator σ then allows the classification of points in a time scale in the following way. If σ(t) > t, then we say that the point t is right scattered; while if σ(t) = t then, we say the point t is right dense.
Throughout this work, the assumption is made that T has the topology that it inherits from the standard topology on the real numbers R. Definition 1.3. Fix t ∈ T and let x : T → R n . Define x ∆ (t) to be the vector (if it exists) with the property that given > 0, there is a neighborhood U of t with It is said that x ∆ (t) is the (delta) derivative of x(t) and that x is (delta) differentiable at t.
, t ∈ T, then it is said that G is a (delta) antiderivative of g and the Cauchy (delta) integral is defined by For a more general definition of the delta integral, see [2,3].
The following theorem is due to Hilger [5].
Theorem 1.5. Assume that g : T → R n and let t ∈ T.
(i) If g is differentiable at t, then g is continuous at t.
(ii) If g is continuous at t and t is right scattered, then g is differentiable at t with (1.8) A. C. Peterson and Y. N. Raffoul 135 (iii) If g is differentiable and t is right dense, then We assume throughout that t 0 ≥ 0 and t 0 ∈ T. By the time scale interval [t 0 ,∞), we mean the set {t ∈ T : t 0 ≤ t < ∞}. The theory of time scales dates back to Hilger [5]. The monographs [2,3,6] also provide an excellent introduction.

Lyapunov functions
In this section, we define what Peterson and Tisdell [7] call a type I Lyapunov function and summarize a few of the results and examples given in [7] relative to what we do here.
Peterson and Tisdell [7] proved that if V is a type I Lyapunov function and the functioṅ V is defined byV where ∇ = (∂/∂x 1 ,...,∂/∂x n ) is the gradient operator and the "·" denotes the usual scalar product, then, if x is a solution to (1.1), it follows that Peterson and Tisdell [7] also show thaṫ Sometimes the domain of V will be a subset D of R n . Note that V = V (x) and even if the vector field associated with the dynamic equation is autonomous,V still depends on t (and x of course) when the graininess function of T is nonconstant. Several formulas are given in Peterson and Tisdell [7] forV (t,x) for various type I Lyapunov functions V (x). In this paper, the only one of these formulas that we will use is that if V (x) = x 2 , theṅ It is the second term in (2.5) that usually makes the Lyapunov theory for time scales much more difficult than the continuous case.

Exponential stability
In this section, we present some results on the exponential stability of the trivial solution of (1.1). First we give a few more preliminaries.
Definition 3.1. Assume that g : T → R. Define and denote g ∈ C rd (T;R) as right-dense continuous (rd-continuous) if g is continuous at every right-dense point t ∈ T and lim s→t − g(s) exists and is finite at every left-dense point t ∈ T, where left-dense is defined in the obvious manner.
If g ∈ C rd , then g has a (delta) antiderivative [2, Theorem 1.74]. Now define the socalled set of regressive functions, by Under the addition on defined by Then define the set of positively regressive functions by For p ∈ , the generalized exponential function e p (·,t 0 ) on a time scale T can be defined (see [2,Theorem 2.35]) to be the unique solution to the IVP We will frequently use the fact that if p ∈ + , then [2, Theorem 2.48] e p (t,t 0 ) > 0 for t ∈ T. We will use many of the properties of this (generalized) exponential function, which are summarized in the following theorem (see [2, Theorem 2.36]). (iv) e p (t,s) = 1/e p (s,t) = e p (s,t); (v) e p (t,s)e p (s,r) = e p (t,r); (vi) e p (t,s)e q (t,s) = e p⊕q (t,s); (vii) e p (t,s)/e q (t,s) = e p q (t,s), where p q := p ⊕ ( q).
It follows from Bernoulli's inequality (see [2,Theorem 6.2]) that for any time scale, if the constant λ ∈ + , then In particular, if T = R, then e λ (t,t 0 ) = e −λ(t−t0) and if T = Z + , then e λ (t,t 0 ) = (1 + λ) −(t−t0) . For the growth of generalized exponential functions on time scales, see Bodine and Lutz [1]. With all this in mind, we make the following definition.
x t,t 0 ,x 0 ≤ C x 0 ,t 0 e M t,t 0 d , ∀t ∈ t 0 ,∞ , (3.8) where · denotes the Euclidean norm on R n . The trivial solution of (1.1) is said to be uniformly exponentially stable on [0,∞) if C is independent of t 0 .
We are now ready to present some results.
Theorem 3.4. Assume that D ⊂ R n contains the origin and there exists a type I Lyapunov  Integrating both sides from t 0 to t with x 0 = x(t 0 ), we obtain, for t ∈ [t 0 ,∞), (3.14) It follows that for t ∈ [t 0 ,∞), Thus by (3.9), This concludes the proof.
We now provide a special case of Theorem 3.4 for certain functions φ and ψ.
Proof. As in the proof of Theorem 3.4, let x be a solution to (1.1)-(1.2) that stays in D for all t ≥ 0. Since M = inf t≥0 λ 3 (t)/[λ 2 (t)] r/q > 0, e M (t,0) is well defined and positive. Since 3) and the product rule Integrating both sides from t 0 to t with x 0 = x(t 0 ), and by invoking condition (3.17) and the fact that λ 1 (t) ≥ λ 1 (t 0 ), we obtain This concludes the proof.

Examples
We now present some examples to illustrate the theory developed in Section 3.