Hamiltonian Formalism for a Multicriteria Optimal Motion Control Problem

Abstract

Statements of and solution methods for dynamic multicriteria optimization problems are considered. Although such problems are usually solved by reduction to the optimization of a scalar function of the criteria, in real-world vector problems one needs to introduce the Pareto frontier and describe its evolution. We propose an approach based on vector dynamic programming and similar to the classical approach. The method involves finding an extremum with respect to the Pareto ordering. A vector value function (the Pareto frontier) is introduced, for which an analog of the optimality principle is stated and the corresponding system of equations of the Hamilton-Jacobi-Bellman type is constructed. The control is sought in the form of synthesis. A method for constructing a guaranteed point estimate of the Pareto frontier is described, and solutions of problems of management by objectives obtained with the use of vector dynamic programming are presented.