2018-Sustainable Industrial Processing Summit
SIPS2018 Volume 8. Composite, Ceramic, Nanomaterials and Mathematics

Editors:F. Kongoli, M. de Campos
Publisher:Flogen Star OUTREACH
Publication Year:2018
Pages:184 pages
ISBN:978-1-987820-96-6
ISSN:2291-1227 (Metals and Materials Processing in a Clean Environment Series)
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    Numerical Solution of Optimal Control Problem for Linear Differential Equations with m-Point Nonlocal Boundary Conditions

    David Devadze1; Vakhtang Beridze1;
    1BATUMI SHOTA RUSTAVELI STATE UNIVERSITY, Batumi, Georgia;
    Type of Paper: Regular
    Id Paper: 281
    Topic: 38

    Abstract:

    Nonlocal boundary value problems are quite an interesting generalization of classical problems, and at the same time, they are naturally obtained when constructing mathematical models of real processes in physics, engineering, and so on [1]. The Bitsadze-Samarski nonlocal boundary value problem [2] arose in connection with mathematical modeling of processes occurring in plasma physics. Nonlocal boundary value problems for quasilinear differential equations of first order on the plane were considered in the work [3]. An m-point nonlocal boundary value problem for generalized analytic functions is formulated in the works [4-5], where the investigation is carried out by the method of reducing nonlocal boundary value problems to a sequence of Riemann-Hilbert problems. When dealing with questions of optimization for systems with distributed parameters, an important tool is the use of problems of the existence of a generalized solution under discontinuous right-hand parts of the equation. In the work [6], a unified scheme is formulated for proving necessary conditions of optimality for a wide class of problems of optimization of objects with distributed parameters. The extension of the maximum principle to nonlocal boundary value problems can be found in [7], while in [8-11] it is shown that that their numerical solutions are of nontrivial nature. The present paper is dedicated to problems of optimal control, whose behavior is described by linear differential equations of first order on the plane with m-point nonlocal boundary conditions. Necessary and sufficient conditions of optimality are obtained. A theorem on the existence and uniqueness of a generalized solution of the conjugate problem is proved. A numerical algorithm of the solution of an optimal control problem is given.

    Keywords:

    Mathematics; Optimal Control; Nonlocal Problems

    References:

    [1] V. V. Shelukhin. A non-local in time model for radionuclides propagation in Stokes fluid. Dinamika Sploshn. Sredy No. 107 (1993), 180-193, 203, 207.<br />[2] A. V. Bitsadze and A. A. Samarskii, Some elementary generalizations of linear elliptic boundary value problems. (Russian) Dokl. Akad. Nauk SSSR 185 (1969), 739-740.<br />[3] V. Beridze, D. Devadze and H. Meladze, On one nonlocal boundary value problem for quasilinear differential equations. Proc. A. Razmadze Math. Inst. 165 (2014), 31-39.<br />[4] M. Abashidze, V. Beridze and D. Devadze, Algorithm for generalized solution of nonlocal boundary value problem. Bull. Georgian Natl. Acad. Sci. 11 (2017), no. 3, 43-48.<br />[5] D. Devadze, The Existence of a Generalized Solution of an m-Point Nonlocal Boundary Value Problem. Commun. Math. 25 (2017), no. 2, 159-169.<br />[6] V. I. Plotnikov, Necessary optimality conditions for controllable systems of a general form. (Russian) Dokl. Akad. Nauk SSSR 199 (1971), 275-278.<br />[7] L. S. Pontryagin, V. G. Boltyanskii, R. V. Gamkrelidze and E. F. Mishchenko, The Mathematical Theory of Optimal Processes. (Russian) Fourth edition. Nauka, Moscow, 1983.<br />[8] D. Sh. Devadze and V. Sh. Beridze, Optimality conditions for quasilinear differential equations with nonlocal boundary conditions. (Russian) Uspekhi Mat. Nauk 68 (2013), no. 4(412), 179-180; translation in Russian Math. Surveys 68 (2013), no. 4, 773-775.<br />[9] D. Devadze and V. Beridze, An optimal control problem for Helmholtz equations with Bitsadze-Samarski boundary conditions. Proc. A. Razmadze Math. Inst. 161 (2013), 47-53.<br />[10] D. Devadze and V. Beridze, An algorithm of the solution of an optimal control problem for a nonlocal problem. Bull. Georgian Natl. Acad. Sci. (N.S.) 7 (2013), no. 1, 44-48.<br />[11] D. Devadze and V. Beridze, Optimality conditions and solution algorithms of optimal control problems for nonlocal boundary-value problems. J. Math. Sci. (N.Y.) 218 (2016), no. 6, 731-736.

    Cite this article as:

    Devadze D and Beridze V. (2018). Numerical Solution of Optimal Control Problem for Linear Differential Equations with m-Point Nonlocal Boundary Conditions. In F. Kongoli, M. de Campos (Eds.), Sustainable Industrial Processing Summit SIPS2018 Volume 8. Composite, Ceramic, Nanomaterials and Mathematics (pp. 183-184). Montreal, Canada: FLOGEN Star Outreach