A method for solving differential inclusions with fixed right end

A. V. Fominykh

Vestnik Sankt-Peterburgskogo universiteta. Prikladnaâ matematika, informatika, processy upravleniâ, Tome 14 (2018) no. 4, pp. 302-315

Résumé

In the paper, we study a differential inclusion with a given continuous convex multivalued mapping. For a given finite time interval, it is required to construct a solution of the differential inclusion, that satisfies the given initial condition or both the initial and final conditions. With the help of support functions, the original problem is reduced to the problem of global minimization of some functions in the space of piecewise continuous functions. In the case of continuous differentiability of the support function of a multivalued mapping with respect to the phase variables, this functional is Gateaux differentiable. In the paper, Gateaux gradient is found, necessary and (in some particular cases) sufficient conditions for the global minimum of the given functions are obtained. On the basis of these conditions, the method of steepest descent is applied to the original problem. Numerical examples illustrate the method realization.

Détail

Keywords: differential inclusions, support function, the steepest descent method.

Bibliographie

[1] Aubin J.-P., Cellina A., Differential inclusions, Springer-Verlag Publ., Berlin, 1984, 344 pp. | MR | Zbl

[2] Polovinkin E. S., Multivalued analysis and differential inclusions, Fizmatlit Publ., M., 2014, 597 pp. (In Russian)

[3] Blagodatskih V. I., Introduction to optimal control, Vysshaya shkola Publ., M., 2001, 239 pp. (In Russian)

[4] Blagodatskih V. I., Filippov A. F., “Differential inclusions and optimal control”, Proceedings of the Steklov Institute of Mathematics, 169 (1985), 194–252 (In Russian) | MR

[5] Filippov A. F., Differential equations with discontinuous right-hand side, Nauka Publ., M., 1985, 226 pp. (In Russian) | MR | Zbl

[6] Watbled F., “On singular perturbations for differential inclusions on the infinite interval”, Journal of Mathematical Analysis and Applications, 310:2 (2005), 362–378 | DOI | MR | Zbl

[7] Gama R., Smirnov G., Watbled F., “Stability and optimality of solutions to differential inclusions via averaging method”, Set-Valued and Variational Analysis, 22:2 (2014), 349–374 | DOI | MR | Zbl

[8] Chengi Yi., “Existence of solutions for a class of nonlinear evolution inclusions with nonlocal conditions”, Journal of Optimization Theory and Applications, 162:1 (2014), 13–33 | DOI | MR

[9] Sandberg M., “Convergence of the forward Euler method for nonconvex differential inclusions”, SIAM J. Numer. Anal., 47:1 (2008), 308–320 | DOI | MR

[10] Bastien J., “Convergence order of implicit Euler numerical scheme for maximal monotone differential inclusions”, Z. Angew. Math. Phys., 64 (2013), 955–966 | DOI | MR | Zbl

[11] Beyn W.-J., Rieger J., “The implicit Euler scheme for one-sided Lipschitz differential inclusions”, Discrete and Continuous Dynamical Systems. Series B, 14:2 (2010), 409–428 | DOI | MR | Zbl

[12] Lempio F., “Modified Euler methods for differential inclusions”, Set-Valued Analysis and Differential Inclusions, A Collection of Papers resulting from a Workshop held in Pamporovo (Bulgaria, September 17–21, 1990), Progr. Systems Control Theory, 16, eds. A. B. Kurzhanski, V. M. Veliov, Birkhauser Verlag Publ., Boston–Basel–Berlin, 1993, 131–148 | MR | Zbl

[13] Veliov V., “Second order discrete approximations to strongly convex differential inclusions”, Systems Control Lett., 13 (1989), 263–269 | DOI | MR | Zbl

[14] Taubert K., “Dierenzenverfahren fiir Schwingungen mit trockener und zdher Reibung und fiir Regelungssysteme”, Numer. Math., 26 (1976), 379–395 | DOI | MR | Zbl

[15] Nikol'skiy M. S., “On approximation of the attainability domain of control process”, Mat. notes, 41 (1987), 71–76 (In Russian)

[16] Nikol'skiy M. S., “On a method for approximation of attainable set for a differential inclusion”, Journal of Vych. Mat. Math. Phys., 28 (1988), 1252–1254 (In Russian)

[17] Panasyuk A. I., “Equations of attainable set dynamics. Pt 1. Integral funnel equations”, Journal of Optim. Theory Appl., 64 (1990), 349–366 | DOI | MR | Zbl

[18] Baier R., Gerdts M., Xausa I., “Approximation of reachable sets using optimal control algorithms”, Numer. Algebra Control Optim., 3 (2013), 519–548 | DOI | MR | Zbl

[19] Dontchev A., Lempio F., “Difference methods for differential inclusions: A survey”, SIAM Rev., 34:2 (1992), 263–294 | DOI | MR | Zbl

[20] Schilling K., “An algorithm to solve boundary value problems for differential inclusions and applications in optimal control”, Numer. Funct. Anal. and Optimiz., 10:7 (1989), 733–764 | DOI | MR | Zbl

[21] Bonnans J. F., Shapiro A., Perturbation analysis of optimization problems, Springer Science+Business Media, New York, 2000, 600 pp. | MR | Zbl

[22] Demyanov V. F., Extremum conditions and variation calculus, Vysshaya shkola Publ., M., 2005, 335 pp. (In Russian)

[23] Kantorovich L. V., Akilov G. P., Functional analysis, Nauka Publ., M., 1977, 741 pp. (In Russian) | MR