Uniform global stability of switched nonlinear systems in the Koopman operator framework

Switched nonlinear systems in the Koopman operator framework: Toward a Lie-algebraic condition for uniform stability

“…We now present an intermediate result that we will use to prove our main stability results. It is adapted from [14], where a switched system was considered instead of (1).…”

“…Under a contraction assumption on the flow ϕ t , one can prove the boundedness and the strong continuity of the Koopman semigroup. In this work, we focus on the evolution of the evaluation functionals k z of the Hardy space (see [14] for the technical details), so that the above properties are not required.…”

“…Moreover, most of our results could be extended to upper triangular Jacobian matrices, a property which is always satisfied in C n×n up to a linear change of coordinates (Schur's theorem). See [14] for this general case.…”

“…where D is the number of upper off-diagonal nonzero entries of JF (0). See the proof of Corollary 3.9 in [14] for more details.…”

“…exponential stability, diagonalizable linear system), the specific case of polynomial vector fields is considered, along with more general analytic vector fields. Our theoretical findings are built upon our previous work based on the properties of the Koopman generator defined in the Hardy space of the polydisc [14]. But in contrast to previous work, they do not focus on switched nonlinear systems and provide stability conditions which are less conservative thanks to the use of re-scaled Hardy spaces.…”