ℒ2-stability for a class of nonlinear systems via potential-based realizations

This paper considers the problem of representing a sufficiently smooth control affine system as a structured potential-driven system and to exploit the obtained representation to study ℒ₂-stability and stabilization. The representation problem has been studied extensively in recent years for particular classes of potential-driven systems, however exploiting these structures, for example generalized Hamiltonian systems, to study input-output stability was not fully investigated in the literature. The present note proposes a geometric decomposition technique, based on the Hodge decomposition theorem, to reexpress a given vector field into a potential-driven form. Using the proposed decomposition technique, finite gain stability conditions are developed, in the form of Hamilton-Jacobi inequalities, based on the convexity of a computed potential.