A max-plus approach to the approximation of transient bounds for systems with nonlinear L2-gain

The notion of nonlinear L2-gain is a natural generalization of the extensively studied conventional (linear) L2-gain property that finds application in stability analysis and H∞-control for nonlinear systems. As in the conventional formulation, notions of transient and gain bounds play an integral role in the statement of the property as an input / output inequality. These bounds summarize an imposed decoupling of system behaviour into transient and asymptotic parts, each of which are important in understanding and quantifying system performance. In this work, a variational approach to the characterization of transient bounds in the presence of a fixed nonlinear gain is considered. Based on an associated dynamic programming principle for this variational characterization, a max-plus eigenvector method for approximating tight transient bounds in the presence of a nonlinear L2-gain bound is considered. Convergence of the associated power method is considered in some detail, whilst it is shown that significant issues remain to be addressed in the approximation of the dynamic programming evolution operators associated with the attendant finite horizon optimization problems.