Some Results Of Stability For Nonlinear Impulsive Differential Systems

In this paper, we study the stability properties of the three impulsive differential systems as follows:(â… ) Nonlinear impulsive control systemwhere f ∈ C[R₊×Rⁿ× R^m, Rⁿ], I_k ∈ C[Rⁿ × R^m, Rⁿ], u is any admissible control vector in given admissible control set Ω, t₀ < t₁ < t₂ < … < t_k < ∝ are impulse times, and t_k →∞, k→∞ .(â…¡) Impulsive hybrid differential systemwhere f ∈ C[R₊ × Rⁿ × R^m, Rⁿ), I_k ∈ C[Rⁿ,Rⁿ], λ_k ∈ C[Rⁿ, R^m), t₀ < t₁ < t₂ < … < t_k < ∞ are impulse times, and t_k → ∞, k → ∞. (â…¢) Impulsive delay differential systemwhere f ∈ C[R₊ × Rⁿ × Rⁿ, Rⁿ], I_k ∈ C[Rⁿ, Rⁿ], t - Ï„(t) ≥ 0, Ï„(t) ≥ 0, 0 ≤ t₀ ≤ Ï„₁ < Ï„₂ < … < Ï„_k < ∞ are impulse times, and Ï„_k → ∞, k →∞.We get the results of stability and boundedness in terms of two measures for system (â… ), the results of strict stability of the trivial solution for system (â…¡), and the results of stability in terms of two measures for system (â…¢). Examples are also discussed to illustrate the theorems, respectively.Impulsive control problem has attracted the interest of many researchers in recent years. Such control arises in a wide variety of applications, such as orbital transfer of satallite, optimal control of nerve network, and control of money supply in a financial market. There are many cases where impulsive control and continuous control can give preferable performance by supplyment each other. In the control theory, continuous control is shown by the fact that there exists an admissible control vector, which satisfies certain conditions, at the right of system. Impulsive control problems are well described by impulsive control systems. At present, the stability theory of nonlinear impulsive control systems has not yet been fully developed, and the known results are established only by virtue of comparison principle of Lyapunov functions'5^6)'t8^33', not employing other methods. In chapter one, we study stability, practical stability and boundedness properties of impulsive control/system (I) in terms of two measures using variational Lyapunov method'7! based on comparison principle, and establish new variational comparison principle and some comparison results. Differently, we consider an ordinary differential system which contains no control, and look impulsive control system (I) as perturbation of the ordinary differential system, and, the Lyapunov function in the paper contains the solution of the ordinary differential system. By using these theorems, we can conclude stsbility properties of impulsive control system (I) from the corresponding stability properties of the relevant ordinary differential system and comparison system. As a result of using variational Lyapunov method, the known results can be looked as special cases of our results.Impulsive hybrid system is called impulsive differential system with variable structure, which describes a great many physical models in applications, and there has been much research about the system in recent yearst10^11^31]''33]. As we know, Lyapunov stability of the trivial solution of a differential system does not rule out the possibility of asymptotic stability. Moreover, the asymptotic stability of the trivial solution only implies that the nontrivial solutions near the trivial solution tend to zero, and it does not guarantee any information about the rate of decay of the solutions which shows how they tend to zero. In other words, these definitions of stability are one-sided estimates for solutions, and they are not strict. So it is natural to expect that an estimation of lower bound for the rate at which solutions approach to the trivial solution would be offered. Such concepts are called stability in tube-like domain or strict stability. Currently, the research about the strict stability for differential systems especially for impulsive systemsis not yet much'12J''13-l'l14^ In chapter two, we investigate the strict stability of impulsive hybrid system (II) employing Lyapunov direct method and comparison principle respectively and get some sufficient conditions.In recent years, the qualitative properties of impulsive delay differential systems have been developed by a large number of mathematicians, and there are some corresponding results for impulsive differential systems with finite delay[18H26l. in the known literatures, the delay of system is always restricted by a given bound, and thus there appears a corresponding method for the research of stability, and some asymptotic stability theorems. However, stability theory of impulsive differential systems with time-varying delay has not yet been fully developed. In chapter three, we investigate the stability in terms of two measures for impulsive differential system (III) with time-varying delay, using Lyapunov function and Razumikhin techniques, and establish some sufficient conditions. Since the delay r(t) of the system is not much restricted, and it could tend to oo, the Razumikhin conditions in our theorems are different from the known ones. Moreover, the theorems in chapter three are about uniform stability in the most, and one cannot get the conclusion about asymptotic stability for system (III) using the known method. In addition, we establish a comparison principle with respect to system (III) in chapter three, and employing the principle, we get a comparison criteria of stability in terms of two measures for the system.