  In this paper, we provide a novel solution to an open problem on the global
uniform stability of switched nonlinear systems. Our results are based on the
Koopman operator approach and, to our knowledge, this is the first theoretical
contribution to an open problem within that framework. By focusing on the
adjoint of the Koopman generator in the Hardy space on the polydisk (or on the
real hypercube), we define equivalent linear (but infinite-dimensional)
switched systems and we construct a common Lyapunov functional for those
systems, under a solvability condition of the Lie algebra generated by the
linearized vector fields. A common Lyapunov function for the original switched
nonlinear systems is derived from the Lyapunov functional by exploiting the
reproducing kernel property of the Hardy space. The Lyapunov function is shown
to converge in a bounded region of the state space, which proves global uniform
stability of specific switched nonlinear systems on bounded invariant sets.
