Some Differential Inequalities on Time Scales and Their Applications to Feedback Control Systems

This paper deals with feedback control systems on time scales. Firstly, we generalize the semicycle concept to time scales and then establish some differential inequalities on time scales. Secondly, as applications of these inequalities, we study the uniform ultimate boundedness of solutions of these systems. We give a new method to investigate the permanence of ecosystem on time scales. And some known results have been generalized. Finally, an example is given to support the result.


Introduction
The theory of time scales was introduced by Hilger in his Ph.D. thesis [1] in 1988.The study of dynamic equations on time scales has recently attracted a lot of attention, because it reveals many discrepancies and helps to avoid proving results twice, for differential equations and difference equations, respectively.The time scales calculus has wide applications in various fields like biology, engineering, economics, physics, neural networks, social sciences, and so on [2].
On the other hand, the permanence of a system is very important.As we know, permanence is affected by such factors as environment, food supplies, attack rates, and so on.As an example, Schreiber and Patel [3] showed that ecoevolutionary feedbacks can mediate permanence at intermediate trade-offs in the attack rates.So it is worthwhile considering the permanence of biological models.The permanence of biological systems has been discussed by many authors [4][5][6][7][8][9][10][11][12][13][14][15][16][17].In the process of studying these problems, the comparison method plays a paramount role.Such as in [4,6], some sufficient conditions of permanence for biological systems were established by the comparison method.Permanence is also applied in physics, chemistry, physiology, diseases, economics, and so on.Such as in [18], sufficient conditions of the permanence for a SEIRS system were obtained.
. ., ) are all positively regressive (R is defined as in the second section (Definition 12)), in which they used the notation where  is a bounded function in T.
They first give the definition of the permanence of ecosystem (1), and the definition of the permanence of other ecosystems mentioned in this paper is similar.
If T = R, one refers the readers to [14,17].Chen and Xie [14] considered the following system: where   (),   (), and   () are all nonnegative continuous functions such that and It is easy to see that the integrations have no influence on the permanence of system (5).They cancelled the additional limitations of [17] and obtained the following.Theorem 3. Assume that (H1) holds.Then system ( 5) is permanent.
If T = Z, one refers the readers to [15,16].Let   () = exp{  ()}; they discussed the discrete form of cooperative system (1): In [16], Li and Zhang used the techniques in [11] instead of the comparison theorem in [15] and obtained the following.
Obviously, in Theorem 2, if T = R (or T = Z), the conditions are stronger than that in Theorem 3 (or Theorem 4).This paper deals with system (1) with the initial condition and one assumes that −  , −  , −  ∈ T, for  ∈ T, which implies that system (1) is valid; sup T = ∞, which ensures the ultimate boundedness of the solutions; and there exists a positive constant  such that one can see that (8) includes the condition: T is  -periodic (in [13]).In fact, if T is -periodic, then sup{() |  ∈ T} ≤ .
Our aim is to unify Theorems 3 and 4 completely by cancelling the additional limitations of Theorem 2.
For simplicity, one first considers a special case (logistic system) of system (1) with the initial condition In this continuous case, one refers the readers to [7,19,20].They considered the system with feedback control in periodic case or boundedness case and obtained permanence, stability, and existence of periodic solutions for the system.
Let () = exp{()}; if T = Z, then system (9) is reformulated as this system or its other forms attracted much interest; one refers the readers to [10][11][12]21].They also considered this system in periodic case or boundedness case and obtained permanence and existence of periodic solutions for this system.Chen [10] discussed the permanence of system (12).Using the comparison theorem, he obtained the following.
But for system (12) itself, (H4) may not be necessary.Fan and Wang [11] used a method instead of the comparison theorem and obtained the following.Theorem 6. Assume (H3) holds.Then system (12) is permanent.
One generalizes the method of semicycle [11] to time scales instead of the comparison theorem.By establishing some differential inequalities on time scales, one proves that solutions of systems (1) and ( 9) are uniformly ultimate bounded.Our result is a unification and extension of Theorems 3 and 4.
The remainder of this paper is arranged as follows.In Section 2, we give some basic properties about time scales and generalize the semicycle concept to time scales.Two differential inequalities on time scales are proved in Section 3. Section 4 is devoted to uniform ultimate boundedness of solutions of systems (1) and ( 9).An example is presented in Section 5 to support the result.Conclusions are presented in Section 6.

Preliminaries
First we give some properties about time scales before our main result (see [1,2]).
Definition 7 (see [1,2]).A time scale T is an arbitrary nonempty closed subset of the real number R. Definition 8 (see [1,2]).For  ∈ T we define the forward jump operator  : T → T and the backward jump operator  : T → T, by Throughout the paper we make an assumption that  ≤ , ,  are points in T.
Definition 9 (see [2]).Define the interval [, ] in T by and as usual, we assume  ∈ T, for all  > ,  ∈ T is a constant.
Definition 12 (see [2]).We say that a function  : T → R is regressive provided holds, where throughout the paper The set of all regressive and rd-continuous functions  : T → R will be denoted in this paper by Definition 13 (see [2]).If  ∈ R, we define the exponential function by where the cylinder transformation  ℎ () = (1/ℎ)log(1 + ℎ), for ℎ > 0.
Similar to the definition of semicycle in discrete case [22], one gives the following.Definition 16.Let  be a constant and : T → R; a positive semicycle relative to  of  is defined as follows: it consists of a "string" of terms all greater than or equal to ; and a negative semicycle relative to  of  is defined as follows: it is a "string" of terms all less than or equal to .

Differential Inequalities
By using the semicycle concept and the comparison theorem, we give some differential inequalities as follows.
(ii) The proof is similar to (i) of Lemma 17.

Applications
Before giving our main result, we list the definition of uniform ultimate boundedness.
Definition 19.The solution ( 1 (),  2 ()) T of system ( 9) is said to be uniformly ultimate bounded if there exist two constants  1 and  2 such that, for any initial condition ( where  1 and  2 are independent on ( The definition of the uniform ultimate boundedness of solutions of -species system is similar.
Then we give the applications of these differential inequalities in this section.
Therefore, the proof of Theorem 21 is completed.
Noticing the differences of Definitions 1 and 19, due to the exponential transformations in Section 1, when solutions of system (1) or ( 9) are uniformly ultimate bounded, systems ( 5) and ( 6) or (11) and ( 12) are permanent, and the contrary also holds true.Then we have the following remarks.If T = R, we could get a new method to solve the continuous result from Theorem 20.Thus, our result is a well unification of continuous and discrete cases.
Remark 23.If T = R, then the result implies Theorem 3.
It is obvious that the conditions of Theorem 21 are weaker than that in Theorem 2, and it includes the known results of [14,16] on R or Z, respectively.Thus, Theorem 21 is a well unification of Theorems 3 and 4.And our result shows that the additional conditions in [15,17] are not necessary.

An Example
In this section, we give a numerical example to support our main result.
here  = 1, 2, 3, . ... By applying numerical analysis from Matlab, we obtain Figure 1.It is obvious that the solutions for system (79) are uniformly ultimate bounded.Our numerical simulation supports our theoretical findings (see Figure 1).

Conclusion
In this paper, we consider two systems with feedback control on time scales.It is shown that these systems contain the differential and difference systems as special cases.By constructing some differential inequalities on time scales, the uniform ultimate boundedness of solutions for these systems is obtained.Compared with some previous relative results, our results could unify the continuous and discrete cases sufficiently (see Theorems 3 and 4).This provides a new