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We study the Hamilton-Jacobi equation for undiscounted exit time control problems with general nonnegative Lagrangians using the dynamic programming approach. We prove theorems characterizing the value function as the unique…
This work extends weak KAM theory to the case of a nonsmooth Lagrangian satisfying a superlinear growth condition. Using the solution of a weak KAM equation that is a stationary Hamilton-Jacobi equation and the proximal aiming method, we…
Let a control system and a target be given on an open subset of an Euclidean space. The existence of a Control Lyapunov Function - namely a positive definite, semiconcave, solution of the Hamilton-Jacobi inequality corresponding to the…
We consider Hamiltonians associated to optimal control problems for affine systems on the torus. They are not coercive and are possibly unbounded from below in the direction of the drift of the system. The main assumption is the strong…
The first-order optimality conditions for a generic nonlinear optimization problem are generated as part of the terminal transversality conditions of an optimal control problem. It is shown that the Lagrangian of the optimization problem is…
A classical problem in ergodic continuous time control consists of studying the limit behavior of the optimal value of a discounted cost functional with infinite horizon as the discount factor $\lambda$ tends to zero. In the literature,…
We study global optimization of non-convex functions through optimal control theory. Our main result establishes that (quasi-)optimal trajectories of a discounted control problem converge globally and practically asymptotically to the set…
The literature on continuous-time stochastic optimal control seldom deals with the case of discrete state spaces. In this paper, we provide a general framework for the optimal control of continuous-time Markov chains on finite graphs. In…
We consider the optimal control of solutions of first order Hamilton-Jacobi equations, where the Hamiltonian is convex with linear growth. This models the problem of steering the propagation of a front by constructing an obstacle. We prove…
We consider an optimal control problem with ergodic (long term average) reward for a McKean-Vlasov dynamics, where the coefficients of a controlled stochastic differential equation depend on the marginal law of the solution. Starting from…
In this article we study ergodic problems in the whole space $\mathbb{R}^N$ for weakly coupled systems of viscous Hamilton-Jacobi equations with coercive right-hand sides. The Hamiltonians are assumed to have a fairly general structure and…
Weak KAM theory for discount Hamilton-Jacobi equations and corresponding discount Lagrangian/Hamiltonian dynamics is developed. Then it is applied to error estimates for viscosity solutions in the vanishing discount process. The main…
At the core of optimal control theory is the Pontryagin maximum principle - the celebrated first order necessary optimality condition - whose solutions are called extremals and which are obtained through a function called Hamiltonian, akin…
We develop the counterpart of weak KAM theory for potential mean field games. This allows to describe the long time behavior of time-dependent potential mean field game systems. Our main result is the existence of a limit, as time tends to…
Main objects of the paper are stationary and weak KAM Hamilton-Jacobi equations on the finite-dimensional torus. The key idea of the paper is to replace the underlying calculus of variations problems with continuous time Markov decision…
This paper is concerned with the asymptotic analysis of infinite systems of weakly coupled stationary Hamilton-Jacobi-Bellman equations as the discount factor tends to zero. With a specific Hamiltonian, we show the convergence of the…
We establish existence and uniqueness of minimax solutions for a fairly general class of path-dependent Hamilton-Jacobi equations. In particular, the relevant Hamiltonians can contain the solution and they only need to be measurable with…
In this paper, we generalize weak KAM theorem from positive Lagrangian systems to "proper" Hamilton-Jacobi equations. We introduce an implicitly defined solution semigroup of evolutionary Hamilton-Jacobi equations. By exploring the…
We study the time-periodic version of Evans approach to weak KAM theory. Evans minimization problem is equivalent to a first oder mean field game system. For the mechanical Hamiltonian we prove the existence of smooth solutions. We…
We consider the homogenization of Hamilton-Jacobi equations and degenerate Bellman equations in stationary, ergodic, unbounded environments. We prove that, as the microscopic scale tends to zero, the equation averages to a deterministic…