arXiv subjects

Alexander Fominyh

Publications and source records attributed to Alexander Fominyh.

5 recordsLinked to original sources

arXiv2024

Method for finding solution to "quasidifferentiable" differential inclusion

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.

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arXiv2023

On differential inclusions arising from some discontinuous systems

The paper deals with systems of ordinary differential equations containing in the right-hand side controls which are discontinuous in phase variables. These controls cause the occurrence of sliding modes. If one uses one of the well-known definitions of the solution of discontinuous systems, then the motion of an object while being on some surface can be described in terms of differential inclusions. With the help of the previously developed apparatus for solving differential inclusions, a method is constructed for finding the trajectories of a system moving in a such a mode. Since some of frequently used discontinuous controls contain nonsmooth functions of phase variables, the paper pays special attention to study the differential properties of such systems. At the end of the paper controls of a slightly different, in contrast to the classical, type are considered which have useful differential properties, and a method is constructed for solving systems with such controls considered both before hitting the required surface and moving in its vicinity.

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arXiv2023

A method for finding a solution to the nonsmooth differential inclusion of a special structure

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.

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arXiv2022

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

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).

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arXiv2022

The Subdifferential Descent Method in a Nonsmooth Variational Problem

The paper is devoted to the classical variational problem with a nonsmooth integrand of the functional to be minimized. The integrand is supposed to be subdifferentiable. Under some natural conditions the subdifferentiability of the functional considered is proved. The problem of finding the subdifferential descent is being solved and the subdifferential descent method is applied to solve the original problem. The algorithm developed is demonstrated by examples.

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