A convex variational principle for the necessary conditions of classical optimal control

TL;DR

This paper introduces a convex variational framework for deriving necessary conditions in classical optimal control, enabling solutions for nonlinear problems without relying on traditional Riccati equations.

Contribution

It develops a family of convex variational principles whose Euler-Lagrange equations correspond to optimal control conditions, including explicit solutions for quadratic regulator problems.

Findings

Explicit functional form for Quadratic-Quadratic Regulator problem

Existence of minimizers rigorously established

Linear-Quadratic Regulator with time-dependent forcing solved without nonlinear methods

Abstract

A scheme for generating a family of convex variational principles is developed, the Euler- Lagrange equations of each member of the family formally corresponding to the necessary conditions of optimal control of a given system of ordinary differential equations (ODE) in a well-defined sense. The scheme is applied to the Quadratic-Quadratic Regulator problem for which an explicit form of the functional is derived, and existence of minimizers of the variational principle is rigorously shown. It is shown that the Linear-Quadratic Regulator problem with time-dependent forcing can be solved within the formalism without requiring any nonlinear considerations, in contrast to the use of a Riccati system in the classical methodology. Our work demonstrates a pathway for solving nonlinear control problems via convex optimization.