A soothing invisible hand: moderation potentials in optimal control

TL;DR

This paper introduces moderation potentials in optimal control, providing a new way to incorporate control effort incentives that are smoothly integrated into Hamiltonian systems, with practical calculations for affine systems.

Contribution

It defines moderation potentials and incentives, linking them to Hamiltonian solutions, and presents a family of such incentives with simple, computable forms for affine systems.

Findings

A family of moderation incentives with explicit potentials is developed.

Controls approach those from logarithmic penalties as parameters vary.

The moderation cost remains bounded and smoothly influences control strategies.

Abstract

A moderation incentive is a continuously differentiable control-dependent cost term that is identically zero on the boundary of the admissible control region, and is subtracted from the `do or die' cost function to reward sub-maximal control utilization in optimal control systems. A moderation potential is a function on the cotangent bundle of the state space such that the solutions of Hamilton's equations satisfying appropriate boundary conditions are solutions of the synthesis problem - the control-parametrized Hamiltonian system central to Pontryagin's Maximum Principle. A multi-parameter family of moderation incentives for affinely controlled systems with quadratic control constraints possesses simple, readily calculated moderation potentials. One member of this family is a shifted version of the kinetic energy-style control cost term frequently used in geometric optimal control. The controls determined by this family approach those determined by a logarithmic penalty function as one of the parameters approaches zero, while the cost term itself is bounded.