Maximum Principle for variational problems with scalar argument

TL;DR

This paper derives a general maximum principle for a broad class of variational problems with various constraints, providing necessary optimality conditions applicable to complex systems including delays and multiple equations.

Contribution

It introduces a unified approach to necessary optimality conditions using a Lagrange function for diverse variational problems with multiple constraint types.

Findings

Necessary conditions involve a Lagrange multipliers vector function.

Optimal solutions maximize the Lagrange function with respect to control variables.

Conditions extend to sliding regimes with infinite switching frequency.

Abstract

In this paper the necessary conditions of optimality in the form of maximum principle are derived for a very general class of variational problems. This class includes problems with any optimization criteria and constraints that can be constructed by combining some basic types (differential equation, integral equations, algebraic equation, differential equations with delays, etc). For each problem from this class the necessary optimality conditions are produced by constructing its Lagrange function $R$ and then by dividing its variables into three groups denoted as $u(t)$, $x(t)$ and $a$ correspondingly. $a$ are parameters which are constant over time. The conditions of optimality state that a non-zero vector function of Lagrange multipliers exists such that on the optimal solution function $R$ attains maximum on $u$, is stationary on $x$, and the integral of $R$ over the control period $S$ can't be improved locally. Similar conditions are also obtained for sliding regimes. Here solution is given by the limit of maximizing sequence on which the variables of the second group are switching with infinite frequency between some (basic) values.