Study On Qualitative Theory Of The One-dimensional State-dependent Switched Systems

Switched systems as an important class of hybrid systems consists of a series of continuous or discrete subsystem an a switching law.It has been widely concerned and studied by mathematician,computer scientists and engineers because of its profound the-oretical significance and wide engineering application.According to the switching law,switch systems can be divided into the switched systems with fixed switching time(open-loop switching systems)and the switched systems with switching law depending on state(close-loop switching systems).In recent years,the research results of switched systems mainly focus on the time-dependent switched systems,and the state-dependent switched system has not been further studied because of its complexity.But the latter is very important in the intelligence times.So in this paper we will mainly study a class of the qualitative theory of the differential switched systems with the switching time depending on state.For the class of linear on-off switched systems with switching law depending on state x(t)= ax(t)+ f?(t),t>t0,x(t0)?x0 and the switching law S = {(ti,f?)|x(ti)= y?,i=1,2,...}.Firstly,by reasonably introduce the weak solution,we obtain the existence and uniqueness of the whole solution in a weaker sense.And then,by introducing new topological structures we prove the solution is continuous dependence and differentiability of the original state.Moreover the derivative is the solution of the differential systems with impulses at variable times.On the basis of the study about linear on-off switched systems with switching time depending on state,we will study the class of nonlinear on-off switched systems with switching time depending on state x(t)?f?(t,x(t)),t>t0,x(t0)= x0 and the switching lawS = {(ti,f?)|x(ti)y?,i=1,2...}.We obtain the existence and uniqueness of the solution and the continuous dependence and differentiability of the solution with respect to the original state.In this paper,we found the switching law can destroy the intrinsic property of the systems,such as the global existence of solution;even in very strong conditions(such as f?)?Cn[t0,+?)),there is no guarantee that solution is continuous dependence of the original state.Moreover the solution is not continuous dependence of the original state in L1.Obvious,the solution is not differentiable of' the original state.So we need to introduce a new topological structures and study the continuous dependence and d-ifferentiability of the solution with respect to the original state in the new topological structures.It will reveal the similarities and differences with the classical switched sys-tems,differential systems.Not only that,the results and ideas in this paper lay the foundation for solving the qualitative theory and optimal control problem of the state-dependent switched systems.