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This paper is devoted to new derivative criterion of metric regularity for set-valued map from a Fréchet-Montel space to a Fréchet space. Such type of criteria have been well studied in Banach spaces. Our work extends recent studies motivated by Nash-Moser-Ekeland approach. As a consequence of our criterion, an implicit mapping result is also given in the paper. [ABSTRACT FROM AUTHOR]

We 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. [ABSTRACT FROM PUBLISHER]

A classical problem in ergodic control consists of studying the limit behavior of the optimal value Vλ of a discounted cost functional with infinite horizon as the discount factor λ tends to zero. In the literature, this problem has been addressed under various conditions ensuring that the rescaled value function λVλ converges uniformly to a constant limit. The main goal of this paper is to study this problem without such conditions, so that the aforementioned limit need not be constant. So, under a nonexpansivity assumption, we derive Lipschitz bounds which yield compactness of {λVλ} for both control systems and differential games. Then, we study the convergence of solutions to Hamilton--Jacobi equations under the hypothesis that the Hamiltonian is radially nondecreasing, hence allowing for the existence noncoercivity directions. Using PDE methods, we show that the convergence is monotone and we characterize the limit as the maximal subsolution of a certain Hamilton--Jacobi equation. Finally, we return to optimal control problems and differential games for nonexpansive dynamics obtaining explicit representation formulas for the uniform limit of λVλ as λ → 0. [ABSTRACT FROM AUTHOR]

This paper studies stability properties of the solutions of optimal control problems for linear systems. The analysis is based on an adapted concept of metric regularity, the strong bi-metric regularity, which is introduced and investigated in the paper. It allows one to give a more precise description of the effect of perturbations on the optimal solutions in terms of a Hölder- type estimate and to investigate the robustness of this estimate. The Hölder exponent depends on a natural number k, which is known as the controllability index of the reference solution. An inverse function theorem for strongly bi-metrically regular mappings is obtained, which is used in the case k = 1 for proving stability of the solution of the considered optimal control problem under small nonlinear perturbations. Moreover, a new stability result with respect to perturbations in the matrices of the system is proved in the general case k ≥ 1. [ABSTRACT FROM AUTHOR]

We study a classical stochastic optimal control problem with constraints and discounted payoff in an infinite horizon setting. The main result of the present paper lies in the fact that this optimal control problem is shown to have the same value as a linear optimization problem stated on some appropriate space of probability measures. This enables one to derive a dual formulation that appears to be strongly connected to the notion of (viscosity sub) solution to a suitable Hamilton-Jacobi-Bellman equation. We also discuss relation with long-time average problems. [ABSTRACT FROM AUTHOR]

This survey paper presents some new advances in theoretical aspects of differential game theory. We particular focus on three topics: differential games with state constraints; backward stochastic differential equations approach to stochastic differential games; differential games with incomplete information. We also address some recent development in nonzero-sum differential games (analysis of systems of Hamilton-Jacobi equations by conservation laws methods; differential games with a large number of players, i.e., mean-field games) and long-time average of zero-sum differential games. [ABSTRACT FROM AUTHOR]

We study a zero-sum differential game where the players have only an unperfect information on the state of the system. In the beginning of the game only a random distribution on the initial state is available. The main result of the paper is the existence of the value obtained through an uniqueness result for Hamilton-Jacobi-Isaacs equations stated on the space of measure in ℝn. This result is the first step for future work on differential games with lack of information. [ABSTRACT FROM AUTHOR]

In this paper, we study the dynamics of a differential inclusion built upon a nonsmooth, not necessarily convex, constrained minimization problem in finite-dimensional spaces. In particular, we are interested in the investigation of the asymptotic behavior of the trajectories of the dynamical system represented by the differential inclusion. Under suitable assumptions on the constraint set and the two involved functions (one defining the constraint set, the other representing the functional to be minimized), it is proved that all the trajectories converge to the set of the constrained critical points. We present also a large class of constraint sets satisfying our assumptions. As a simple consequence. in the case of a smooth convex minimization problem, we have that any trajectory converges to the set of minimizers. [ABSTRACT FROM AUTHOR]

Presents a study that aimed to give a representation formula for the mean curvature motion (MCM) in terms of the value function of some stochastic optimal control problem. Distance solution of the MCM; Discussion on the study of the distance solutions; Generalization of the results to general geometric equations.

Abstract. We characterize functions satisfying a dissipative inequality associated with a control problem. Such a characterization is provided in terms of an epicontingent solution, or a viscosity supersolution to a partial differential equation called Isaacs' equation. Links between supersolutions and epicontingent solutions to Isaacs' equation are studied. Finally, we derive (possibly discontinuous) disturbance attenuation feedback of the H[sup∞] problem from contingent formulation of Isaacs' equation. [ABSTRACT FROM AUTHOR]

We study optimal times to reach a given closed target for controlled systems with a state constraint. Our goal is to characterize these optimal time functions in such a way that it is possible to compute them numerically and we do not need to compute trajectories of the controlled systems. In this paper we provide new results using viability theory. This allows us to study optimal time functions free from the controllability assumptions classically made in the partial differential equations approach. [ABSTRACT FROM AUTHOR]

We study control problems with several targets in the case of nonlinear dynamic systems. The map associating with every initial condition the minimal time to reach successively two given targets is characterized in the framework of differential inclusions through the notion of viability kernel. This approach allows one to treat the problem without assumptions of regularity and to build numerical schemes computing the minimal time. We also study the problem where an order of visit of the targets is required. The statements are also extended to the case of p targets under state constraints. Equivalent formulations in terms of Hamilton–Jacobi equations are also provided. [ABSTRACT FROM AUTHOR]