Geodesic
Control and perturbation in Sturm — Liouville's problem with discontinuous nonlinearity
Vestnik Sankt-Peterburgskogo universiteta. Prikladnaâ matematika, informatika, processy upravleniâ, Tome 19 (2023) no. 2, pp. 275-282
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We consider the Sturm — Liouville problem with discontinuous nonlinearity, control and perturbation. Previously obtained results for equations with a spectral parameter and a discontinuous operator are applied to this problem. By the variational method, we have established theorems on the existence of solutions to the Sturm — Liouville problem with discontinuous nonlinearity and to the optimal control problem, as well as on topological properties of the set of the acceptable “control — state” pairs. A one-dimensional analog of the Gol'dshtik model for separated flows of an incompressible fluid with control and perturbation is given as an application.
Mots-clés : Sturm — Liouville's problem
Keywords: discontinuous nonlinearity, control problems, variational method, Gol'dshtik's model. 
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O. V. Baskov; D. K. Potapov. Control and perturbation in Sturm — Liouville's problem with discontinuous nonlinearity. Vestnik Sankt-Peterburgskogo universiteta. Prikladnaâ matematika, informatika, processy upravleniâ, Tome 19 (2023) no. 2, pp. 275-282. http://geodesic.mathdoc.fr/item/VSPUI_2023_19_2_a11/

[1] Potapov D. K.

[2] Proceedings of the Steklov Institute of Mathematics, 17, no. 1, 2011, 190–200 (In Russian)

[3] Potapov D. K., “On resolvability of a control problem for one class of equations with discontinuous operators and a spectral parameter”, Proceedings of Voronezh State University. Series Systems Analysis and Information Technologies, 2011, no. 2, 36–39 (In Russian)

[4] Potapov D. K., “Control problems for higher-order systems of elliptic type with a spectral parameter, an external perturbation, and a discontinuous nonlinearity”, Bulletin of Voronezh State Technical University, 8:1 (2012), 55–57 (In Russian)

[5] Potapov D. K., “Control problems for equations with a spectral parameter and a discontinuous operator under perturbations”, Journal of Siberian Federal University. Series Mathematics and Physics, 5:2 (2012), 239–245 (In Russian) | MR | Zbl

[6] Potapov D. K., “Existence of solution to control problems with perturbations for a class of equations with spectral parameter and discontinuous operator”, Proceedings of Voronezh State University. Series Systems Analysis and Information Technologies, 2012, no. 1, 12–15 (In Russian)

[7] Potapov D. K., “On dependence between control and state in spectral problems for equations with discontinuous operators”, Bulletin of Voronezh State Technical University, 9:5–1 (2013), 104–105 (In Russian)

[8] Potapov D. K., “Optimal control of higher order elliptic distributed systems with a spectral parameter and discontinuous nonlinearity”, Journal of Computer and Systems Sciences International, 2013, no. 2, 19–24 (In Russian) | DOI | Zbl

[9] Budak B. M., Berkovich E. M., “Optimal control problems for differential equations with discontinuous right sides”, USSR Computational Mathematics and Mathematical Physics, 11:1 (1971), 51–64 (In Russian) | Zbl

[10] Carl S., Heikkila S., “On the existence of minimal and maximal solutions of discontinuous functional Sturm — Liouville boundary value problems”, J. Inequal. Appl., 2005, no. 4, 403–412 | MR | Zbl

[11] Bonanno G., Bisci G. M., “Infinitely many solutions for a boundary value problem with discontinuous nonlinearities”, Bound. Value Probl, 2009, 670675, 20 pp. | DOI | MR | Zbl

[12] Bonanno G., Buccellato S. M., “Two point boundary value problems for the Sturm — Liouville equation with highly discontinuous nonlinearities”, Taiwanese J. Math, 14:5 (2010), 2059–2072 | DOI | MR | Zbl

[13] Potapov D. K., “Sturm — Liouville's problem with discontinuous nonlinearity”, Differential Equations, 50:9 (2014), 1284–1286 (In Russian) | DOI | Zbl

[14] Potapov D. K., “Existence of solutions, estimates for the differential operator, and a “separating” set in a boundary value problem for a second-order differential equation with a discontinuous nonlinearity”, Differential Equations, 51:7 (2015), 970–974 (In Russian) | DOI | Zbl

[15] Bonanno G., D'Agui G., Winkert P., “Sturm — Liouville equations involving discontinuous nonlinearities”, Minimax Theory Appl, 1:1 (2016), 125–143 | MR | Zbl

[16] Pavlenko V. N., Postnikova E. Yu., “Sturm — Liouville problem for an equation with a discontinuous nonlinearity”, Chelyabinsk Physical and Mathematical Journal, 4:2 (2019), 142–154 (In Russian) | MR | Zbl

[17] Lions Zh.-L., Control of distributed singular system, Nauka Publ, M., 1987, 368 pp. (In Russian)

[18] Chang K.-C., “Free boundary problems and the set-valued mappings”, Journal of Differential Equations, 49:1 (1983), 1–28 | DOI | MR | Zbl

[19] Pavlenko V. N., Potapov D. K., “Existence of a ray of eigenvalue for equations with discontinuous operators”, Siberian Mathematical Journal, 42:4 (2001), 911–919 (In Russian) | MR | Zbl

[20] Potapov D. K., “Mathematical model for separated flows of incompressible fluid”, Proceedings of RANS. Series Mathematics. Mathematical Modeling. Informatics and Control, 8:3–4 (2004), 163–170 (In Russian)

[21] Potapov D. K., “Continuous approximations for a 1D analog of the Gol'dshtik model for separated flows of an incompressible fluid”, Numerical Analysis and Applications, 14:3 (2011), 291–296 (In Russian) | MR | Zbl