Method for finding solution to "quasidifferentiable" differential inclusion

TL;DR

This paper introduces a method for solving a special class of quasidifferentiable differential inclusions by reducing the problem to a variational form, proving quasidifferentiability, and applying a modified steepest descent method with convergence analysis.

Contribution

It develops a novel approach to handle quasidifferentiable differential inclusions, including reduction to variational problems and applying a quasidifferential descent method.

Findings

The functional for the problem is quasidifferentiable.

Necessary minimum conditions are formulated using quasidifferential.

Numerical examples demonstrate the method's effectiveness.

Abstract

The paper explores the differential inclusion of a special form. It is supposed that the support function of the set in the right-hand side of an inclusion may contain the sum of the maximum and the minimum of the finite number of continuously differentiable (in phase coordinates) functions. It is required to find a trajectory that would satisfy differential inclusion with the boundary conditions prescribed and simultaneously lie on the surface given. We give substantial examples of problems where such differential inclusions may occur: models of discontinuous systems, linear control systems where the control function or/and disturbance of the right-hand side is/are known to be subject to some nonsmooth (in phase vector) constraints, some real mechanical models and differential inclusions per se with special geometrical structure of the right-hand side. The initial problem is reduced to a variational one. It is proved that the resulting functional to be minimized is quasidifferentiable. The necessary minimum conditions in terms of quasidifferential are formulated. The (modified) steepest (or the quasidifferential) descent method in a classical form is then applied to find stationary points of the functional obtained. Herewith, the functional is constructed in such a way that one can verify whether the stationary point constructed is indeed a global minimum point of the problem. The ``weak'' convergence of the method proposed is proved for some particular cases. The method constructed is illustrated by numerical examples.