Search

We characterize functions satisfying a dissipative inequality associated with a control problem. Such a characterization is provided in terms of epicontingent and viscosity supersolutions to a Partial Differential Equation called the Hamilton-Jacobi-Bellman-Isaacs equation. Links between viscosity and epicontingent supersolutions are studied. Finally, we derive (possibly discontinuous) disturbance attenuation feedback of the H^{infty}-problem from contingent formulation of the Isaacs' Equation.

We study convergence of value-functions associated to control systems with a singular perturbation. In the nonlinear case, we prove new convergence results: the limit of optimal costs of the perturbed system is an optimal cost for the reduced system. We furthermore provide an estimation of the rate of convergence when the reduced system has solutions regular enough.

In this paper, we study two new methods for approximating the viability kernel of a given set for a Holderian differential inclusion. We approximate this kernel by viability kernels for discrete dynamical systems. We prove a convergence result when the differential inclusion is replaced by a sequence of recursive inclusions. Furthermore, when the given set is approached by a sequence of suitable finite sets, we prove our second main convergence result. This paper is the first step to obtain numerical methods.

The authors study value functions of a differential game with payoff which depends on the state at a given end time. They consider differential games with feedback strategies and with nonanticipating strategies. They prove that value-functions are solutions to some Hamilton-Jacobi-Isaacs equations in the viscosity and contingent sense. For these two notions of strategies, with some regularity assumptions, The authors prove that value-functions are the unique solution of Isaacs' equations.

The authors investigate a differential inclusion whose solutions have to remain in a given closed set. The viability kernel is the set of the initial conditions starting at which, there exist solutions to the differential inclusion remaining in this closed set. In this paper, the authors provide an algorithm which determine this set and they apply it to some concrete examples.

The author studies where and how solutions associated to a differential inclusion can or cannot enter a given target. For this purpose, he associates partitions of the target boundary with the dynamics of the system. He qualitatively describes the behavior of these solutions in terms of viability and invariance kernels of sets. These kernels determine points such that there exist (respectively all) solutions starting at these point remain in a given set of constraints. He also studies the sets which are reached in finite time by viable solutions to the system. Finally, he provides some applications to control systems with one target and he generalizes the concept of semipermeable barrier.