Control problems with differential constraints of higher order

TL;DR

This paper introduces a new optimality principle called 'Principle of Minimal Labour' for control problems constrained by higher order Euler-Lagrange differential equations, extending classical optimal control results.

Contribution

It generalizes the Pontryagin Maximum Principle to higher order differential constraints without fixed initial data, based on a novel variational principle.

Findings

Established the 'Principle of Minimal Labour' for higher order control problems.

Derived a generalized Pontryagin Maximum Principle under new conditions.

Proved the equivalence between optimal controls and a variational inequality along homotopies.

Abstract

We consider cost minimising control problems, in which the dynamical system is constrained by higher order differential equations of Euler-Lagrange type. Following ideas from a previous paper by the first and the third author, we prove that a curve of controls $u_o(t)$ and a set of initial conditions $\sigma_o$ gives an optimal solution for a control problem of the considered type if and only if an appropriate double integral is greater than or equal to zero along any homotopy $(u(t, s), \sigma(s))$ of control curves and initial data starting from $u_o(t) = u(t, 0)$ and $\sigma_o = \sigma(0)$. This property is called "Principle of Minimal Labour". From this principle we derive a generalisation of the classical Pontryagin Maximum Principle that holds under higher order differential constraints of Euler-Lagrange type and without the hypothesis of fixed initial data.