Christian Mugisho Zagabe (University of Namur)

Title: Lie-algebraic condition for uniform stability of switched nonlinear systems: Construction of a common Lyapunov function by the Koopman operator approach.

Abstract: Switched systems consist of a finite set of dynamical systems and a switching signal indicating which system is activated. Their interest is due to the fact that many phenomena in nature, engineering,… are represented by different models depending on the change of some parameters. Their stability properties are not intuitive. Indeed, for two individually stable (or unstable) systems one can construct a signal that makes the whole system unstable (or stable). In the uniform stability theory, the main goal is to find sufficient (and/or necessary) conditions that make the whole system stable for all possible switching laws.

In 1999, Daniel Liberzon and co-workers proved that for a switched linear system, the solvability property of the Lie algebra generated by Hurwitz matrices is a sufficient condition for uniform stability.

In 2004, Daniel Liberzon formulated an open problem to find which condition of the Lie algebra (of vector fields) can be used to guarantee the global uniform asymptotic stability (GUAS) of switched nonlinear systems. 

Using the Koopman operator approach, we propose an answer to the open Liberzon problem. Our result is related to the dynamics on the polydisc and shows that a sufficient condition for GUAS follows from the solvability of the Lie algebra generated by the Hurwitz Jacobian matrices of the vector fields.

More precisely, we construct a common Lyapunov function for switched nonlinear systems, which is convergent in a specific region of the state space. This is done by defining the Koopman operator on the Hardy space on the polydisc where the reproducing kernel property allows us to obtain a Lyapunov function via the evolution of the evaluation functional. We then infer the GUAS property of switched nonlinear systems on a specific invariant set.

The seminar will take place in room S08 at the Faculty of Sciences