Nonstandard duality and nonlocal necessary optimality conditions in nonconvex optimal control problems

• Published in: Automation and remote control Vol. 75; no. 11; pp. 1906 - 1921
• Main Author: Dykhta, V. A.
• Format: Journal Article
• Language: English
• Published: Moscow Pleiades Publishing 01.11.2014
• Subjects: CAE) and Design; Calculus of Variations and Optimal Control; Optimization; Computer-Aided Engineering (CAD; Control; Control systems; Mathematical analysis; Mathematics; Mathematics and Statistics; Maximum principle; Mechanical Engineering; Mechatronics; Nonlinear Systems; Optimal control; Optimization; Problem solving; Remote control; Robotics; Systems Theory; Trajectories; Dual Problem; Optimal Control Problem; Maximum Principle; Remote Control; Minimum Principle
• ISSN: 0005-1179; 1608-3032
• DOI: 10.1134/S0005117914110022

To partially implement the idea of considering nonlinear optimal control problems immediately on the set of Pontryagin extremals (or on quasiextremals if the optimal solution does not exist), we introduce auxiliary functions of canonical variables, which we call bipositional, and the corresponding modified Lagrangian for the problem. The Lagrangian is subject to minimization on the trajectories of the canonical system from the Maximum Principle. This general approach is further specialized for nonconvex problems that are linear in state, leading to a nonstandard dual optimal control problem on the trajectories of the adjoint system. Applying the feedback minimum principle to both original and dual problems, we have obtained a pair of necessary optimality conditions that significantly strengthen the Maximum Principle and admit a constructive realization in the form of an iterative problem solving procedure. The general approach, optimality features, and the iterative solution procedure are illustrated by a series of examples.