Bilateral Fixed-Points and Algebraic Properties of Viability Kernels and Capture Basins of Sets

Abstract

Many concepts of viability theory such as viability or invariance kernels and capture or absorption basins under discrete multivalued systems, differential inclusions and dynamical games share algebraic properties that provide simple – yet powerful – characterizations as either largest or smallest fixed points or unique minimax (or bilateral fixed-point) of adequate maps defined on pairs of subsets. Further, important algorithms such as the Saint-Pierre viability kernel algorithm for computing viability kernels under discrete system and the Cardaliaguet algorithm for characterizing ‘discriminating kernels’ under dynamical games are algebraic in nature. The Matheron Theorem as well as the Galois transform find applications in the field of control and dynamical games allowing us to clarify concepts and simplify proofs.