A max-plus affine power method for approximation of a class of mixed L2 / L∞ value functions

This paper is concerned with the approximation via max-plus methods of value functions arising in a class of mixed L₂ / L_∞ optimization problems. The class of problems is defined to be suitably general so as to admit future application to L₂-gain analysis, L_∞ bounded (LIB) dissipation and to the analysis of systems with the input to state stability (ISS) property. Common to each of these problems is the applicability of dynamic programming, which naturally leads to the formulation of max-plus methods using the resulting semigroups (and sub-semigroups). This paper provides the details of this formulation. In particular, we develop an affine power method that yields the correct solution of the dynamic programming principle (DPP) and hence the underlying optimization problem, despite the inherent non-uniqueness of solutions of such DPPs.