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We investigate Hamilton Jacobi Isaacs equations associated to a two-players zero-sum differential game with incomplete information. The first player has complete information on the initial state of the game while the second player has only information of a – possibly uncountable – probabilistic nature: he knows a probability measure on the initial state. Such differential games with finite type incomplete information can be viewed as a generalization of the famous Aumann–Maschler theory for repeated games. The main goal and novelty of the present work consists in obtaining and investigating a Hamilton Jacobi Isaacs Equation satisfied by the upper and the lower values of the game. Since we obtain a uniqueness result for such Hamilton Jacobi equation, as a byproduct, this gives an alternative proof of the existence of a value of the differential game (which has been already obtained in the literature by different technics). Since the Hamilton Jacobi equation is naturally stated in the space of probability measures, we use the Wasserstein distance and some tools of optimal transport theory. [ABSTRACT FROM AUTHOR]

We investigate an optimal control problem with an averaging cost. The asymptotic behaviour of the values is a classical problem in ergodic control. To study the long run averaging we consider both Cesàro and Abel means. A main result of the paper says that there is at most one possible accumulation point – in the uniform convergence topology – of the values, when the time horizon of the Cesàro means converges to infinity or the discount factor of the Abel means converges to zero. This unique accumulation point is explicitly described by representation formulas involving probability measures on the state and control spaces. As a byproduct we obtain the existence of a limit value whenever the Cesàro or Abel values are equicontinuous. Our approach allows to generalise several results in ergodic control, and in particular it allows to cope with cases where the limit value is not constant with respect to the initial condition. [ABSTRACT FROM AUTHOR]

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]

Abstract: We give a representation for the closed convex hull of the set of discounted occupational measures generated by control-state trajectories of a deterministic control system. We also investigate the limit behavior of the latter when the discount factor tends to zero and compare it with the limit behavior of the long run time average occupational measures set. The novelty of our results is in that we allow the control set dependence on the state variables that make the results to be applicable to differential inclusions. [Copyright &y& Elsevier]

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]

We investigate infinite-dimensional differential inclusions. Sufficient conditions for nonemptiness of the viability kernels of such systems are obtained. [Copyright &y& Elsevier]

In the present paper, we prove that the optimal cost of a Mayer''s problem for a control system with singular perturbation does converge to the optimal cost of a Mayer''s problem associated with control system obtained by averaging. The main novelty of our result lies on the fact that we did not require any continuity assumption on the final cost. [Copyright &y& Elsevier]

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]

The problem of compatibility of a stochastic control system and a set of constraints – the so called viability property – has been widely investigated during the last three decades. Given a stochastic control system, the question is to characterize sets A such that for any initial condition in A there exists a control ensuring that the associated stochastic process remains forever almost surely in A (this is called the viability property of A). When A is closed and the dynamics is continuous, the viability property has been characterized in the literature through several equivalent geometric conditions involving A , the drift and the diffusion of the control system. In this article we give a necessary and sufficient condition involving the boundary of an open set A ensuring the viability property of A , whenever A has a C 2 , 1 boundary and the dynamics are Lipschitz. If moreover a classical convexity condition on the control dynamics holds true, we show that the viability of an open set A is equivalent to the viability of its closure. This last result is rather surprising, because several very elementary examples in the deterministic framework show that, in general, there is no such equivalence for a general open set A. We will also discuss examples illustrating that the above equivalence is wrong when either the boundary of A does not have enough regularity, or the dynamics are not Lipschitz continuous. [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]

Abstract: When 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. [Copyright &y& Elsevier]

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.