Abstract
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.
Highlights
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
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
Summary
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. 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. Δy (n) = −a (n) y (n) + b (n) N (n) , n ∈ Z; this system or its other forms attracted much interest; one refers the readers to [10,11,12, 21] They considered this system in periodic case or boundedness case and obtained permanence and existence of periodic solutions for this system.
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