A Numerical Method for Finding the Optimal Solution of a Differential Inclusion

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 maximum of the finite number of continuously differentiable (in phase coordinates) functions. It is required to find a trajectory that would satisfy the differential inclusion with the boundary conditions prescribed and simultaneously lie on the surface given. Such problems arise while practical modeling of discontinuous systems and in other applied problems. The initial problem is reduced to a variational one. It is proved that the resulting functional to be minimized is superdifferentiable. The necessary minimum conditions in terms of superdifferential are formulated. The superdifferential (or the steepest) descent method in a classical form is then applied in order to find stationary points of this functional. Herewith, the functional is constructed in a such a way that one can verify whether the stationary point constructed is indeed a global minimum point of the problem. The convergence of the method proposed is proved. The method constructed is illustrated by examples.

A method for solving discontinuous systems with sliding modes

“…Problem 1. We formulate the problem as follows: it is required to find such a trajectory * ∈ [0, ] (with the derivative ̇ * ∈ [0, ]) which moves along discontinuity surface (9) (the parameters * ∈ are to be determined as well) while ∈ [0, ], satisfies differential inclusion (13) and boundary conditions ( 7), (8). Assume that there exists such a solution.…”