Switched systems became more and more interesting since they are conceptually closed to the description of real complex dynamics in which the state is not necessarily fixed in time but can abruptly change with the environment. In this context, not only a finite number of subsystems (said modes) are given to describe the possible state of the system, but also a switching (or commutation) law is assigned to indicate the active mode at each time.
The stability theory of such systems is not intuitive since it is influenced by the commutation law, which plays a capital role.
This dissertation investigates the uniform stability (i.e. stability under any commutation laws) and the switching stabilization (design of a stabilizing commutation law) problems of switched nonlinear systems.
In the last decades, these problems have mainly been studied for switched linear systems and partially solved for the nonlinear case.
The strategy exploited here is based on a successful tool today: the Koopman operator. This is a linear operator acting on an infinite-dimensional space of functions valued on the nonlinear system's state space. Roughly speaking, it allows one to transform a nonlinear finite-dimensional dynamics into a linear infinite-dimensional dynamics, from which one can deduce results for the original nonlinear system. More precisely, we utilize the Koopman operator framework to address switched nonlinear systems' uniform stability and stabilization problems.
For the first problem, by using a Lie-algebraic solvability condition, we show that individual globally asymptotically stable nonlinear vector fields which admit a common Koopman finite-dimensional invariant subspace generate a uniformly globally asymptotically stable switched nonlinear system. In a broader context, we develop a general framework for studying the (uniform) stability of (switched) nonlinear systems on the polydisk or the hypercube. This systematic approach allows us to construct a common Lyapunov function that guarantee global uniform asymptotic stability on the polydisk or the hypercube. We then apply this framework to derive systematic criteria for the global stability of nonlinear systems defined on the polydisk or the hypercube. Finally, for the second problem, we utilize the previously developed results to provide a state-dependent switching stabilization strategy from a systematic Lyapunov function of a convex combination of nonlinear vector fields.
