Cristopher Hermosilla

Hamilton–Jacobi–Bellman Equations

In this chapter we present recent developments in the theory of Hamilton–Jacobi–Bellman (HJB) equations as well as applications. The intention of this chapter is to exhibit novel methods and techniques introduced few years ago in order to solve long-standing questions in nonlinear optimal control theory of Ordinary Differential Equations (ODEs).

Hamilton–Jacobi–Bellman equations for optimal control processes with convex state constraints

Abstract This work aims at studying some optimal control problems with convex state constraint sets. It is known that for state constrained problems, and when the state constraint set coincides with the closure of its interior, the value function satisfies a Hamilton–Jacobi equation in the constrained viscosity sense. This notion of solution has been introduced by H.M. Soner (1986) and provides a characterization of the value functions in many situations where an inward pointing condition (IPC) is satisfied. Here, we first identify a class of control problems where the constrained viscosity notion is still suitable to characterize the value function without requiring the IPC. Moreover, we generalize the notion of constrained viscosity solutions to some situations where the state constraint set has an empty interior.

Self-dual approximations to fully convex impulsive systems

Fully convex optimal control problems contain a Lagrangian that is jointly convex in the state and velocity variables. Problems of this kind have been widely investigated by Rockafellar and collaborators if the Lagrangian is coercive and without state constraints. A lack of coercivity implies the dual has nontrivial state constraints, and vice versa (that is, they are dual concepts in convex analysis). We consider a framework using Goebels self-dualizing technique that approximates both the primal and dual problem simultaneously and maintains the duality relationship. Previous results are applicable to the approximations, and we investigate the limiting behavior as the approximations approach the original problem. A specific example is worked out in detail.