In this paper, we study a optimal control problem of the value of the solution to an elliptic equation in a bounded domain with a smooth boundary by means of a flow through the domain boundary. We consider the operator of the equation, which is the sum of the Laplace operator with a small coefficient and a zero-order operator. The control is constrained by an integral relation. As a performance index, we employ the sum of the squared norm of the deviation of a state from a prescribed state on the domain boundary and the squared norm of the control. We obtain a complete asymptotic expansion of the solution to the problem in powers of the small parameter.
Keywords: singular problems, optimal control, boundary value problems for systems of partial differential equations, asymptotic expansions
eceived November 21, 2024
Revised January 22, 2025
Accepted January 27, 2025
Aleksey Rufimovich Danilin, Dr. Phys.-Math. Sci., Prof., Krasovskii Institute of Mathematics and Mechanics of the Ural Branch of the Russian Academy of Sciences, Yekaterinburg, 620108 Russia, e-mail: dar@imm.uran.ru
Igor’ Viktorovich Pershin, Krasovskii Institute of Mathematics and Mechanics of the Ural Branch of the Russian Academy of Sciences, Yekaterinburg, 620108 Russia, e-mail: piv@imm.uran.ru
REFERENCES
1. Lions J.L. Controle optimal de systemes gouvernes par des equations aux derivees partielles. Paris, Dunod, Gauthier-Villars, 1968, 426 p. Translated to Russian under the title Optimal’noye upravleniye sistemami, opisyvayemymi uravneniyami s chastnymi proizvodnymi, Moscow, Mir Publ., 1972, 414 p.
2. Danilin A.R. Asymptotic behaviour of bounded controls for a singular elliptic problem in a domain with a small cavity. Sb. Math., 1998, vol. 189, no. 11, pp. 1611–1642. https://doi.org/10.1070/SM1998v189n11ABEH000364
3. Danilin A.R. Approximation of a singularly perturbed elliptic problem of optimal control. Sb. Math., 2000, vol. 191, no. 10, pp. 1421–1431. https://doi.org/10.1070/SM2000v191n10ABEH000512
4. Kapustyan V.E. Asymptotics of bounded controls in optimal elliptic problems. Dokl. Akad. Nauk Ukr., 1992, no. 2, pp. 70–74 (in Russian).
5. Kapustyan V.E. Optimal bisingular elliptic problems with bounded control. Dokl. Akad. Nauk Ukr., 1993, no. 6, pp. 81—85.
6. Danilin A.R., Zorin A.P. Asymptotics of a solution to an optimal boundary control problem. Proc. Steklov Inst. Math., 2010, vol. 269, suppl. 1, pp. S81–S94. https://doi.org/10.1134/S0081543810060088
7. Lou H., Yong J. Second-order necessary conditions for optimal control of semilinear elliptic equations with leading term containing controls. Math. Control Relat. Fields, 2018, vol. 8, no. 1, pp. 57–88. https://doi.org/10.3934/mcrf.2018003
8. Betz Livia M. Second-order sufficient optimality conditions for optimal control of nonsmooth, semilinear parabolic equations. SIAM J. Control Optim., 2019, vol. 57, no. 6, pp. 4033–4062. https://doi.org/10.1137/19M1239106
9. Lubyshev F.V., Fairuzov M.E. Approximation of optimal control problems for semilinear elliptic convection–diffusion equations with boundary observation of the conormal derivative and with controls in coefficients of the convective transport operator and in nonlinear term of the equation. Comput. Math. Math. Phys., 2024, vol. 64, no. 7, pp. 1443–1460. https://doi.org/10.1134/S0965542524700659
10. Danilin A.R. Optimal boundary control in a small concave domain. Ufimsk. Mat. Zh., 2012, vol. 4, no. 2, pp. 87–100 (in Russian).
11. Danilin A.R. Asymptotics of the solution in a problem of optimal boundary control of a flow through a part of the boundary. Proc. Steklov Inst. Math., 2016, vol. 292, suppl. 1, pp. S55–S66. https://doi.org/10.1134/S008154381602005X
12. Il’in A.M. Matching of asymptotic expansions of solutions of boundary value problems. Providence, Amer. Math. Soc., 1992, 281 p. ISBN: 978-0-8218-4561-5. Original Russian text published in Il’in A. M. Soglasovanie asimtoticheskikh razlozhenii reshenii kraevykh zadach, Moscow, Nauka Publ., 1989, 336 p.
13. Vishik M.I., Lyusternik L.A. Regular degeneration and boundary layer for linear differential equations with small parameter. Uspehi Mat. Nauk (N.S.), 1957, vol. 12, no. 5(77), pp. 3–122 (in Russian).
14. Il’in A.M. A boundary layer. Itogi Nauki i Tekhniki. Ser. Sovrem. Probl. Mat. Fund. Napr., 1988, vol. 34, pp. 175–213 (in Russian).
Cite this article as: A.R. Danilin,I.V. Pershin. Asymptotics of a solution to an optimal boundary control problem with performance index defined on a boundary. Trudy Instituta Matematiki i Mekhaniki UrO RAN, 2025, vol. 31, no. 2, pp. 94–107.
