Invariance Envelopes and Invariance Kernels for Lipschitzean Differential Inclusions

The author investigates a differential inclusion whose solutions have to remain in a given closed set. The invariance kernel is the set of the initial conditions starting at which, all solutions to the differential inclusion remain in this closed set. The invariance envelope is the smallest set which contains the given closed set and which is invariant for the differential inclusion. In this paper, the author studies invariance envelopes and he compares this envelope to invariance kernels. He provides an algorithm which determines the invariance kernel and consequently the invariance envelope

Playable Differential Games

Playability conditions of differential games are studied by using Viability Theory. First, the results on playability of time independent differential games are extended to time dependent games. In fact, time is introduced in the dynamics of the game, in the state dependent constraints bearing on controls and in state constraints. Second, some examples of pursuit games are studied. Necessary and sufficient conditions of playability of the game are provided. Here, pursuit games are directly considered as "games of kind" (in Isaacs' sense) and are not considered as "games of degree". The viability condition does not always provide the "optimal strategy" to be as close as possible to a certain goal, but it supplies strategies allowing the system to reach a given goal

Differential Inclusions and Target Problems

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

Viability Kernels of Differential Inclusions with Constraints: Algorithms and Applications

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

Isaacs' Equations for Value-Functions of Differential Games

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

An Algorithm for Viability Kernels in Hoelderian Case: Approximation by Discrete Dynamical Systems

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

Dissipative Control Systems and Disturbance Attenuation for Nonlinear H - Problems

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

Second-order differential equations with random perturbations and small parameters

International audienceWe consider boundary-value problems for differential equations of second order containing a Brownian motion (random perturbation) and a small parameter and prove a special existence and uniqueness theorem for random solutions. We study the asymptotic behaviour of these solutions as the small parameter goes to zero and show the stochastic averaging theorem for such equations. We find the explicit limits for the solutions as the small parameter goes to zero

On limiting values of stochastic differential equations with small noise intensity tending to zero

AbstractWhen the right-hand side of an ordinary differential equation (ODE in short) is not Lipschitz, neither existence nor uniqueness of solutions remain valid. Nevertheless, adding to the differential equation a noise with nondegenerate intensity, we obtain a stochastic differential equation which has pathwise existence and uniqueness property. The goal of this short paper is to compare the limit of solutions to stochastic differential equation obtained by adding a noise of intensity ε to the generalized Filippov notion of solutions to the ODE. It is worth pointing out that our result does not depend on the dimension of the space while several related works in the literature are concerned with the one dimensional case

Singular Perturbations in Non-Linear Optimal Control Systems

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