The paper deals with deterministic optimal control problems with state constraints and non-linear dynamics. It is known for such problems that the value function is in general discontinuous and its characterization by means of a Hamilton-Jacobi equation requires some controllability assumptions involving the dynamics and the set of state constraints. Here, we first adopt the viability point of view and look at the value function as its epigraph. Then, we prove that this epigraph can always be described by an auxiliary optimal control problem free of state constraints, and for which the value function is Lipschitz continuous and can be characterized, without any additional assumptions, as the unique viscosity solution of a Hamilton-Jacobi equation. The idea introduced in this paper bypasses the regularity issues on the value function of the constrained control problem and leads to a constructive way to compute its epigraph by a large panel of numerical schemes. Our approach can be extended to more general control problems. We study in this paper the extension to the infinite horizon problem as well as for the two-player game setting. Finally, an illustrative numerical example is given to show the relevance of the approach.

This paper is concerned with the study of a model case of first order Hamilton-Jacobi equations posed on a “junction”, that is to say the union of a finite number of half-lines with a unique common point. The main result is a comparison principle. We also prove existence and stability of solutions. The two challenging difficulties are the singular geometry of the domain and the discontinuity of the Hamiltonian. As far as discontinuous Hamiltonians are concerned, these results seem to be new. They are applied to the study of some models arising in traffic flows. The techniques developed in the present article provide new powerful tools for the analysis of such problems.

In this paper we prove a comparison result between semicontinuousviscosity subsolutions and supersolutions to Hamilton-Jacobi equations of the form $u_t+H(x,Du) = 0$ in ${\rm I}\!{\rmR}^n\times(0,T)$ where the Hamiltonian H may be noncoercive inthe gradient Du. As a consequence of the comparison result and the Perron's method we get the existence of a continuous solution of this equation.

This errata corrects one error in the 2004 version of this paper [Mennucci, ESAIM: COCV10 (2004) 426–451].

We study the Hamilton-Jacobi equation of the minimal time function in a domain which contains the target set. We generalize the results of Clarke and Nour [J. Convex Anal., 2004], where the target set is taken to be a single point. As an application, we give necessary and sufficient conditions for the existence of solutions to eikonal equations.

We formulate an Hamilton-Jacobi partial differential equation H( x, D u(x))=0 on a n dimensional manifold M, with assumptions of convexity of H(x, .) and regularity of H (locally in a neighborhood of {H=0} in T*M); we define the “minsol solution” u, a generalized solution; to this end, we view T*Mas a symplectic manifold.The definition of “minsol solution” is suited to proving regularity results about u; in particular, we provein the first part that the closure of the set where u is not regular may be covered by a countable number of $n-1$ dimensional manifolds, but for a ${{\mathcal H}}^{n-1}$ negligeable subset.These results can be applied to the cutlocus of a C 2 submanifold of a Finsler manifold.

We investigate the value function of the Bolza problem of the Calculus of Variations
$$ V (t,x)=\inf \left\{ \int_{0}^{t} L (y (s),y' (s))ds + \varphi (y(t)) : y \in W^{1,1} (0,t;\mathbb{R}^n),\; y(0)=x \right\},$$ with a lower semicontinuous Lagrangian L and a final cost $ \varphi $, and show that it is locally Lipschitz for t>0 whenever L is locally bounded. It also satisfies Hamilton-Jacobi inequalities in a generalized sense. When the Lagrangian is continuous, then the value function is the unique lower semicontinuous solution to the corresponding Hamilton-Jacobi equation, while for discontinuous Lagrangian we characterize the value function by using the so called contingent inequalities.