Method of quasidifferential descent in the problem of bringing a nonsmooth system from one point to another

Abstract

The paper considers the problem of constructing program control for an object described by a system with nonsmooth (but only quasidifferentiable) right-hand side. The goal of control is to bring such a system from a given initial position to a given final state in certain finite time. The admissible controls are piecewise continuous and bounded vector-functions with values from some parallelepiped. The original problem is reduced to unconditional minimisation of some penalty functional which takes into account constraints in the form of differential equations, constraints on the initial and the final positions of the object as well as constraints on controls. Moreover, it is known that this functional vanishes on the solution of the original problem and only on it. The quasidifferentiability of this functional is proved, necessary and sufficient conditions for its minimum are written out in terms of quasidifferential. Further, in order to solve the obtained minimisation problem in the functional space the method of quasidifferential descent is applied. The algorithm developed is demonstrated by examples.