A Method of the Quasidifferential Descent in a Problem of Bringing a Nonsmooth System from One Point to Another

TL;DR

This paper introduces a novel quasidifferential descent method for solving nonsmooth optimal control problems, simplifying the structure and enabling pointwise optimality conditions and effective discretization.

Contribution

The paper develops a new technical approach that separates the trajectory and its derivative, simplifying the quasidifferential structure and improving the construction of descent directions.

Findings

The method effectively solves nonsmooth control problems with pointwise optimality conditions.

Discretization after quasidifferential derivation enhances computational efficiency.

Examples demonstrate the method's applicability to nonsmooth Lagrange control problems.

Abstract

The paper considers the problem of constructing program control for an object described by a system with a quasidifferentiable right-hand side. The control aim is to bring the system from a given initial position to a given final state in given finite time. The admissible controls are piecewise continuous vector-functions with values from a parallelepiped. The original problem is reduced to unconditional minimization of a functional. Herewith, the new technical idea is implemented to consider phase trajectory and its derivative as independent variables (and to take the natural relation between them into account via a special penalty function). This idea qualitatively simplified the quasidifferential structure and allowed to overcome the principal difficulties in constructing the steepest descent direction. The quasidifferentiability of the functional is proved, necessary conditions for its minimum are obtained in terms of quasidifferential. In contrast to the existing ones, due to the mentioned idea to ``separate'' the trajectory and its derivative the obtained optimality conditions in the paper are pointwise. In order to solve the obtained minimization problem in the functional space the quasidifferential descent method is applied. Then the discretization is implemented. In contrast to majority of existing methods when the initial problem is discretized, here the discretization is implemented after the quasidifferential is already obtained. The quasidifferential descent directions are calculated independently at each time moment of discretization due to the comparatively simple quasidifferential structure, possible to obtain via the technical idea noted. The algorithm developed is demonstrated by examples. The proposed method can be applied to nonsmooth optimal control problem in Lagrange form (additionally the integral with a quasidifferentiable integrand is to be minimized).