Asymptotic stability of m-switched systems using Lyapunov functions

An investigation of asymptotic stability of m-switched systems based on Lyapunov functions is given. An m-switched system is a system x.=A_ikx where A_ik∈{A₁,. . .,A _m} and i_k is an index from a switching sequence, i.e., control is achieved by switching between possible A _i-matrices. The authors consider questions such as: existence of regions, called Ω-regions, where the m-switched system energy decreases as measured by Lyapunov functions associated with these regions; inclusion of one Ω-region by another; coverage of the state-space by the totality of these regions; and structural conditions on the time derivatives of the Lyapunov functions so that this state-space coverage can be accomplished. Answering these questions is necessary to satisfy a theorem and its associated conditions for asymptotic stability of an m-switched system