On the Relation between the Minimum Principle and Dynamic Programming for Classical and Hybrid Control Systems

TL;DR

This paper explores the relationship between the Hybrid Minimum Principle and Hybrid Dynamic Programming in hybrid control systems, showing they are governed by the same equations under certain conditions, with applications to linear quadratic problems.

Contribution

It establishes the equivalence between the adjoint process and the gradient of the value function in hybrid control, unifying two fundamental optimal control approaches.

Findings

The adjoint process and the value function gradient satisfy the same differential equations.

Under certain assumptions, the Hybrid Minimum Principle and Hybrid Dynamic Programming are equivalent.

A Riccati formalism for linear quadratic hybrid tracking problems is developed.

Abstract

Hybrid optimal control problems are studied for a general class of hybrid systems where autonomous and controlled state jumps are allowed at the switching instants and in addition to terminal and running costs switching between discrete states incurs costs. The statements of the Hybrid Minimum Principle and Hybrid Dynamic Programming are presented in this framework and it is shown that under certain assumptions the adjoint process in the Hybrid Minimum Principle and the gradient of the value function in Hybrid Dynamic Programming are governed by the same set of differential equations and have the same boundary conditions and hence are almost everywhere identical to each other along optimal trajectories. Analytic examples are provided to illustrate the results and, in particular, a Riccati formalism for linear quadratic hybrid tracking problems is presented.