Asymptotics of a solution to a problem of optimal boundary control with two small cosubordinate parameters. II

A. R. Danilin

Trudy Instituta matematiki i mehaniki, Trudy Instituta Matematiki i Mekhaniki UrO RAN, Tome 27 (2021) no. 2, pp. 108-119

Résumé

We consider a problem of optimal boundary control for solutions of an elliptic type equation in a bounded domain with smooth boundary with a small coefficient at the Laplace operator, a small coefficient, cosubordinate with the first, at the boundary condition, and integral constraints on the control: $$ \left\{ \begin{array}{ll} \displaystyle {\mathcal L}_\varepsilon z\mathop{:=}\nolimits - \varepsilon^2 \Delta z + a(x) z = f(x), \displaystyle x\in \Omega,\ \ z \in H^1(\Omega), \\[3ex] \displaystyle l_{\varepsilon} z\mathop{:=}\nolimits \varepsilon^\beta \frac{\partial z}{\partial n} = g(x) + u(x), x\in\Gamma, \end{array} \right. $$ $$ J(u) \mathop{:=}\nolimits \|z-z_d\|^2 + \nu^{-1}|||u|||^2 \to \inf, \quad u \in \mathcal{U}, $$ where $0\varepsilon\ll 1$, $\beta\geqslant 0$, $\beta\in\mathbb{Q}$, $\nu>0,$ $H^1(\Omega)$ is the Sobolev function space, $\partial z/\partial n$ is the derivative of $z$ at the point $x\in\Gamma$ in the direction of the outer (with respect to the domain $\Omega$) normal, $$ \begin{array}{c} \displaystyle a(\cdot), f(\cdot), z_d(\cdot) \in C^\infty(\overline{\Omega}), \quad g(\cdot)\in C^\infty(\Gamma),\quad \forall\, x\in \overline{\Omega}\quad a(x)\geqslant \alpha^2>0, \\[2ex] \displaystyle \mathcal{U} = \mathcal{U}_1,\quad \mathcal{U}_r\mathop{:=}\nolimits \{u(\cdot)\in L_2(\Gamma)\colon |||u||| \leqslant r\}. \end{array} $$ Here $\|\cdot\|$ and $|||\cdot|||$ are the norms in the spaces $L_2(\Omega)$ and $L_2(\Gamma)$, respectively. We find a complete asymptotic expansion of the solution of the problem in powers of the small parameter in the case where $\beta\geqslant 3/2$. In contrast to the previously considered case, the relevance of the constraints on the control depends on $|||g|||$.

Détail

Keywords: singular problems, optimal control, boundary value problems for systems of partial differential equations, asymptotic expansions.

Bibliographie

[1] Lions Zh.-L., Optimalnoe upravlenie sistemami, opisyvaemymi uravneniyami s chastnymi proizvodnymi, Mir, M., 1972, 414 pp.

[2] Danilin A.R., “Asimptotika resheniya zadachi optimalnogo granichnogo upravleniya s dvumya malymi sopodchinennymi parametrami”, Tr. In-ta matematiki i mekhaniki UrO RAN, 26:1 (2020), 102–111 | DOI | MR

[3] Casas E., “A review on sparse solutions in optimal control of partial differential equations”, SeMA J., 74 (2017), 319–344 | DOI | MR | Zbl

[4] Lou H., Yong J., “Second-order necessary conditions for optimal control of semilinear elliptic equations with leading term containing controls”, Math. Control Relat. Fields, 8:1 (2018), 57–88 | DOI | MR | Zbl

[5] Betz Livia M., “Second-order sufficient optimality conditions for optimal control of nonsmooth, semilinear parabolic equations”, SIAM J. Control Optim., 57:6 (2019), 4033–4062 | DOI | MR | Zbl

[6] Kapustyan V.E., “Asimptotika ogranichennykh upravlenii v optimalnykh ellipticheskikh zadachakh”, Dokl. AN Ukrainy. Cer. Matematika, estestvoznanie, tekhnicheskie nauki, 1992, no. 2, 70–74

[7] Danilin A.R., “Optimalnoe granichnoe upravlenie v oblasti s maloi polostyu”, Ufim. mat. zhurn., 4:2 (2012), 87–100 | MR

[8] Sobolev S.L., Nekotorye primeneniya funktsionalnogo analiza v matematicheskoi fizike, Izd-vo LGU, L., 1950, 255 pp. | MR

[9] Lions Zh.-L., Madzhenes E., Neodnorodnye granichnye zadachi i ikh prilozheniya, Mir, M., 1971, 371 pp.

[10] Danilin A.R., “Asimptotika resheniya bisingulyarnoi zadachi optimalnogo granichnogo upravleniya v ogranichennoi oblasti”, Zhurn. vychisl. matematiki i mat. fiziki, 58:11 (2018), 1808–1814

[11] Ilin A.M., Soglasovanie asimptoticheskikh razlozhenii reshenii kraevykh zadach, Nauka, M., 1989, 336 pp.

[12] Vishik M.I., Lyusternik L.A., “Regulyarnoe vyrozhdenie i pogranichnyi sloi dlya lineinykh differentsialnykh uravnenii s malym parametrom”, Uspekhi mat. nauk, 12:5 (1957), 3–122 | Zbl

[13] Ilin A.M., “Pogranichnyi sloi”, Itogi nauki i tekhniki VINITI AN SSSR. Sovremennye problemy matematiki. Fundamentalnye napravleniya, 34, VNITI, M., 1988, 175–214