Topics considered here include: examples of optimal control problems; dynamic programming and the Hamilton-Jacobi-Bellman equation; verification theorems; the Pontryagin Maximum Principle Principle. The examples include many with an economic flavor, but others too (including the Hopf-Lax solution formula for ut + H(Du) = 0 with H convex). There’s much more here than we’ll have time to do in lec...

The main purpose of this paper is to discuss the minimization of energy spent in order that a controlled diffusion process reaches a given target, a d-dimensional bounded domain. The exterior Dirichlet problem for the Hamilton-Jacobi-Bellman equation is studied for a class of criteria which includes the case of energy. Extensions to diffusion with jumps, examples and some other reachability pro...

We apply the stochastic Perron method of Bayraktar and Sîrbu to a general infinite horizon optimal control problem, where the state X is a controlled diffusion process, and the state constraint is described by a closed set. We prove that the value function v is bounded from below (resp., from above) by a viscosity supersolution (resp., subsolution) of the related state constrained problem for t...

We determine the solutions of a nonlinear Hamilton-Jacobi-Bellman equation which arises in the modelling of mean-variance hedging subject to a terminal condition. Firstly we establish those forms of the equation which admit the maximal number of Lie point symmetries and then examine each in turn. We show that the Lie method is only suitable for an equation of maximal symmetry. We indicate the a...

Optimal control problems for a class of nonlinear descriptor systems are considered. It is shown that they possess a well-defined analytical feedback solution in a neighborhood of the origin, provided stabilizability and some other regularity conditions are satisfied. Explicit formulas for the series expansions of the cost function and control law are given.

In Stochastic Optimal Control (SOC) one minimizes the average cost-to-go, that consists of the cost-of-control (amount of efforts), cost-of-space (where one wants the system to be) and the target cost (where one wants the system to arrive), for a system participating in forced and controlled Langevin dynamics. We extend the SOC problem by introducing an additional cost-of-dynamics, characterize...

In this paper, a family of high order numerical methods are designed to solve the Hamilton-Jacobi equation for the viscosity solution. In particular, the methods start with a hyperbolic conservation law system closely related to the Hamilton-Jacobi equation. The compact one-step one-stage Lax-Wendroff type time discretization is then applied together with the local-structure-preserving disconti...

The Hamilton-Jacobi equation for a Hamiltonian section on a Lie affgebroid is introduced and some examples are discussed.

In this work we present a formal generalization of the Hamilton-Jacobi formalism, recently developed for singular systems, to include the case of Lagrangians containing variables which are elements of Berezin algebra. We derive the HamiltonJacobi equation for such systems, analizing the singular case in order to obtain the equations of motion as total differential equations and study the integr...

Hamilton-Jacobi theory for general relativity provides an elegant covariant formulation of the gravitational field. A general ‘coordinate-free’ method of integrating the functional Hamilton-Jacobi equation for gravity and matter is described. This series approximation method represents a large generalization of the spatial gradient expansion that had been employed earlier. Additional solutions ...