| There are many practical examples of impulsive control systems. Three typical examples are the insect population control system whose state variables are number of insects and their natural enemies, a chemical reactor system with the quantities of different chemicals server as states variables, and a financial system with two state variables of amount of money in a market and the saving rates of a central bank. Impulses have an important impact on the stability of the systems. However, the stability is very important to the analysis and application of the system. Therefore, it is very necessary to investigate the stability of impulsive control systems.Systems with impulse effects have been wildly discussed in recent years. A number of stability criteria for such systems were proposed using various approaches. Generally, there are two basic ways. One is using comparison theorem. The other is using Lyapunov functions which are required to be nonincreasing along the whole sequence or subsequence of the switchings. In this paper, the uniform asymptotic stability(UAS) of perturbed dynamical systems with impulse effects is studied by using Lyapunov techniques. A new Lyapunov function is established based on a given Lyapunov function of a nominal stable system. Meanwhile, using comparison theorem the stability is further proved.The controllability is also very important to the systems. It is a significant and fundamental subject in control system theory. As for the controllability of continuous-time linear systems, necessary and sufficient conditions are obtained. But many non-negative systems, such as systems under positive input, are discussed not only controllability, but also positive controllability. It's very important to test the controllability in designing systems. In this paper, positive controllability of continuous-time linear systems is considered. Based on the given theory, a feasible and wildly used algorithm to test the positive controllability is given. |
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