Volterra functional-operator equations and distributed optimization problems

V. I. Sumin

Vestnik rossijskih universitetov. Matematika, Tome 23 (2018) no. 124, pp. 745-756

Résumé

A survey of the results obtained in the theory of optimization of distributed systems by the method of Volterra functional-operator equations is given. Topics are considered: the conditions for preserving the global solvability of controllable initial-boundary value problems, optimality conditions, singular controlled systems in the sense of J.L. Lions, singular optimal controls, numerical optimization methods substantiation and others.

Détail

Keywords: Volterra functional-operator equations, controlled initial-boundary value problems, conditions for preserving global solvability, optimization problems, optimality conditions, optimization methods.

Bibliographie

[1] V. I. Sumin, Optimization of Generalized Controlled Volterra Systems, Diss. dots Cand. Phys.-Math. Sciences, Gorky State University, Gorky, 1975, 158 pp. (In Russian)

[2] V. I. Sumin, “Volterra Functional-operator equations in the theory of optimal control of distributed systems”, Proceedings of the USSR Academy of Sciences, 39:2 (1989), 374–378 (In Russian)

[3] V. I. Sumin, “The Features of Gradient Methods for Distributed Optimal-Control Problems”, Computational Mathematics and Mathematical Physics, 30 (1990), 1-15

[4] V. I. Sumin, “Sufficient conditions for stable existence of solutions to global problems in control theory”, Differential Equations, 26:12 (1990), 1579–1590

[5] V. I. Sumin, Functional Volterra Equations in the Theory of Optimal Control of Distributed Systems, N. I. Lobachevsky State University of Nizhny Novgorod, Nizhny Novgorod, 1992, 110 pp. (In Russian)

[6] V. I. Sumin, Functional Volterra Equations in the Mathematical Theory of Optimal Control of Distributed Systems, Diss. dots Doct. Phys.-Math. Sciences, N. I. Lobachevsky State University of Nizhny Novgorod, Nizhny Novgorod, 1998, 346 pp. (In Russian)

[7] V. I. Sumin, “On problem of singularity of controllable distributed parameter systems. I”, Nizhny Novgorod University Reports. Series: Mathematical Modeling and Optimal Control, 1999, no. 2(21), 145–155 (In Russian)

[8] V. I. Sumin, “On problem of singularity of controllable distributed parameter systems. II”, Nizhny Novgorod University Reports. Series: Mathematical Modeling and Optimal Control, 2001, no. 1(23), 198–204 (In Russian)

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[11] V. I. Sumin, “Conditions of the existence of global solutions of controlled boundary problems for non-linear parabolic equations”, Tambov University Reports. Series: Natural and Technical Sciences, 5:4 (2000), 493–495 (In Russian)

[12] V. I. Sumin, “The problem of sustainability of global solutions existence of controlled boundary problems and Volterra functional equations”, Nizhny Novgorod University Reports. Series: Mathematics yr 2003, no. 1, 91-107 (In Russian)

[13] I. V. Lisachenko, V. I. Sumin, “Nonlinear Goursat-Darboux Control Problem: Conditions for the Preservation of Global Solvability”, Differential Equations yr 2011, 47:6, 858–870 (In Russian)

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[16] A. V. Chernov, Volterra Operator Equations and Their Application to the Hyperbolic Systems Optimization Theory, Diss. dots Cand. Phys.-Math. Sciences, N. I. Lobachevsky State University of Nizhny Novgorod, Nizhny Novgorod yr 2000, 177 pp. (In Russian)

[17] V. I. Sumin, A. V. Chernov, “About sufficient conditions for the existence stability of Volterra operator equations global solutions”, Nizhny Novgorod University Reports. Series: Mathematical Modeling and Optimal Control, 2003, no. 1(26), 39–49 (In Russian)

[18] J. L. Lions, Controle des systemes distribues singuliers, Gauthier-Villars, Paris, 1983

[19] A. F. Filippov, Differential Equations with a Disconnected Right Part, Nauka Publ., Moscow, 1985, 224 pp. (In Russian)

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[22] V. I. Sumin, “Uniform quasinilpotency: definitions, conditions, examples of applications”, Tambov University Reports. Series: Natural and Technical Sciences, 15:1 (2010), 453–466

[23] V. I. Sumin, A. V. Chernov, “Operators in Spaces of Measurable Functions: the Volterra Property and Quasinilpotency”, Differential Equations, 34:10 (1998), 1402–1411 (In Russian)

[24] V. I. Sumin, “On functional Volterra Equations”, Russian Mathematics, 1995, no. 9, 65–75

[25] V. I. Plotnikov, V. I. Sumin, “Optimization of distributed systems in Lebesgue space”, Siberian Mathematical Journal, 22:6 (1981), 913-929

[26] V. I. Plotnikov, “Necessary conditions of optimality for control systems of general type”, Proceedings of the USSR Academy of Sciences, 199:2 (1971), 1069-1073

[27] V. I. Sumin, “Strong degeneration of singular controls in problems of optimization of distributed systems”, Optimization, 1993, no. 52(69), 74–94 (In Russian)

[28] I. V. Lisachenko, V. I. Sumin, “On singular controls of a maximum principle for the problem of the Goursat-Darboux system optimization”, The Bulletin of Udmurt University. Mathematics. Mechanics. Computer Science, 25:4 (2015), 483–491 (In Russian)

[29] I. V. Gorokhova, V. I. Sumin, “About singular controls of pointwise maximum principle for optimization problem connected with Goursat–Darboux system”, Tambov University Reports. Series: Natural and Technical Sciences, 23:122 (2018), 278–284 (In Russian)

[30] F. P. Vasilev, Metody optimizatsii, Izd-vo MTsNMO, M., 2011, 433 pp. (In Russian)

[31] V. I. Sumin, “Strong degeneration of singular controls in distributed optimization problems”, Proceedings of the USSR Academy of Sciences, 320:2 (1991), 295–299 (In Russian)

[32] V. I. Sumin, “On singular controls in the sense of the pointwise maximum principle in distributed optimization problems”, The Bulletin of Udmurt University. Mathematics. Mechanics. Computer Science, 20:3 (2010), 70–80 (In Russian)