This paper is a survey of the Hamilton-Jacobi partial differential equation. We begin with its origins in Hamilton’s formulation of classical mechanics. Next, we show how the equation can fail to have a proper solution. Setting this issue aside temporarily, we move to a problem of optimal control to show another area in which the equation arises naturally. In the final section, we present some ...

We study a class of optimal control problems with state constraints, where the state equation is a differential equation with delays. This class includes some problems arising in economics, in particular the so-called models with time to build, see [1, 2, 26]. We embed the problem in a suitable Hilbert space H and consider the associated Hamilton-Jacobi-Bellman (HJB) equation. This kind of infi...

We present a scheme for the determination of control Lyapunov functions which can be used as a basis for numerical computations. Under the assumption of local asymptotic nullcontrollability we de ne the domain of asymptotic nullcontrollability. On this set a control Lyapunov function is de ned via an optimal control problem. It is then shown that this function can be characterized as the unique...

In a series of papers, we proved theorems characterizing the value function in exit time optimal control as the unique viscosity solution of the corresponding Bellman equation that satisfies appropriate side conditions. The results applied to problems which satisfy a positivity condition on the integral of the Lagrangian. This positive integral condition assigned a positive cost to remaining ou...

This paper provides a characterization of viability kernels and capture basins of a target viable in a constrained subset as a unique closed subset between the target and the constrained subset satisfying tangential conditions or, by duality, normal conditions. It is based on a method devised by Hélène Frankowska for characterizing the value function of an optimal control problem as generalized...

We prove a theorem of existence of time-optimal curves in the space of probability measures, a Dynamic Programming Principle, a controllability result and some comparisons between the classical and the generalized framework. A proper definition of viscosity solution, together with some approximation results, leaded us to prove that the generalization of the minimum time function solves a suitab...

We study the Bellman equation for undiscounted exit time optimal control problems for nonlinear systems and nonnegative instantaneous costs using the dynamic programming approach. Using viscosity solution theory, we prove a uniqueness theorem that characterizes the value functions for these problems as the unique viscosity solutions of the corresponding Bellman equations that satisfy appropriat...

We study an innnite horizon optimal control problem for a system with two state variables. One of them has the evolution governed by a controlled ordinary diierential equation and the other one is related to the latter by a hysteresis relation, represented here by either a play operator or a Prandtl-Ishlinskii operator. By dynamic programming, we derive the corresponding (discontinuous) rst ord...

We review some recent work in fast, efficient and accurate methods to compute viscosity solutions and non-viscosity solutions to static HamiltonJacobi equations which arise in optimal control, anisotropic front propagation, and multiple arrivals in wave propagation. For viscosity solutions, the class of algorithms are known as “Ordered Upwind Methods”, and rely on a systematic ordering inherent...