A variational approach to ℒ∞-gain analysis of nonlinear systems

The L_∞-gain of nonlinear systems is characterized by means of the value function of an associated variational problem. The value function is shown to be the unique continuous viscosity solution of the variational inequality in the finite horizon case, and the minimal lower semicontinuous solution in the infinite horizon case. A condition that guarantees local boundedness of the infinite horizon value function is presented. Approximation schemes for the variational inequalities are developed in the framework of discrete dynamic programming. The control theoretic origin of these inequalities is exploited to develop algorithms for computing upper and lower bounds for their solutions. The framework developed enables computation of the L_∞-induced norm over bounded sets of signals. The only regularity required for the underlying dynamics is continuity, allowing the analysis of a large class of systems including saturated feedback loops