A method for solving differential inclusions with fixed right end

Fominyh, Alexander

doi:10.21638/11701/spbu10.2018.403

“…In paper [12] a method for solving the classical nonsmooth (but only subdifferential) variational problem is proposed based on the idea of considering phase trajectory and its derivative as independent variables (and taking the natural connection between these variables into account via the special penalty term). Reducing solving differential inclusions to a variational problem via corresponding support functions was carried out in papers [13], [14], [15]. Some other nonsmooth problems of optimal control and variational calculus were considered via similar methods in papers [16], [17].…”

Section: Introductionmentioning

confidence: 99%

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

Fominyh

¹

2018

Vestnik St.Petersb. Univ.Math.

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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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“…where instead of the phase variable ( ) one should write its expression via its derivative ( ) by formula (14). Then as previously construct the functional [1], [2] ( , , ) = [1], [2] ( , , ) + ( ),…”

Section: Denotementioning

confidence: 99%

A method for solving discontinuous systems with sliding modes

Fominyh¹

2022

Preprint

0

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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 in a sliding mode 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 sliding 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 discontinuity surface and moving in its vicinity.

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Method for Finding Solution to Nonsmooth Differential Inclusion of Special Structure

Fominyh

2024

ESAIM: COCV

0

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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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A method for solving differential inclusions with fixed right end

Cited by 3 publications

References 7 publications

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

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

A method for solving discontinuous systems with sliding modes

Method for Finding Solution to Nonsmooth Differential Inclusion of Special Structure

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