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 $|||g|||$.
Keywords: singular problems, optimal control, boundary value problems for systems of partial differential equations, asymptotic expansions
Received January 31, 2021
Revised February 10, 2021
Accepted February 15, 2021
Aleksei 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
REFERENCES
1. Lions J.-L. Optimal control of systems governed by partial differential equations. Berlin; N Y: Springer-Verlag, 1971, 396 p. ISBN: 9783540051152 . Translated to Russian under the title Optimal’noe upravlenie sistemami, opisyvaemymi uravneniyami s chastnymi proizvodnymi. Moscow: Mir Publ., 1972, 414 p.
2. Danilin A.R. Asymptotics of a solution to a problem of optimal boundary control with two small cosubordinate parameters. Trudy Inst. Mat. i Mekh. UrO RAN, 2020, vol. 26, no. 1, pp. 102–111 (in Russian). doi: 10.21538/0134-4889-2020-26-1-102-111
3. Casas E. A review on sparse solutions in optimal control of partial differential equations. SeMA J., 2017, vol. 74, pp. 319–344. doi: 10.1007/s40324-017-0121-5
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, 2018, vol. 8, no. 1, pp. 57–88. doi: 10.3934/mcrf.2018003
5. 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. doi: 10.1137/19M1239106
6. Kapustyan V.E. Asymptotics of bounded controls in optimal elliptic problems. Dokl. Akad. Nauk Ukrainy, 1992, no. 2, pp. 70–74 (in Russian).
7. Danilin A.R. Optimal boundary control in a small concave domain. Ufimsk. Mat. Zh., 2012, vol. 4, no. 2, pp. 87–100 (in Russian).
8. Sobolev S.L. Some applications of functional analysis in mathematical physics. Providence, RI: Amer. Math. Soc., 1991, 286 p. ISBN: 0-8218-4549-7 . Original Russian text (1st ed.) published in Sobolev S.L. Nekotorye primeneniya funktsional’nogo analiza v matematicheskoi fizike. Leningrad: Leningr. Gos. Univ. Publ., 1950, 255 p.
9. Lions J.-L., Magenes E. Non-homogeneous boundary value problems and their applications. Berlin: Springer-Verlag, 1972, 357 p. ISBN: 3540053638 . Translated to Russian under the title Neodnorodnye granichnye zadachi i ikh prilozheniya. Moscow: Mir Publ., 1971, 371 p.
10. Danilin A.R. Asymptotics of the solution of a bisingular optimal boundary control problem in a bounded domain. Comput. Math. Math. Phys., 2018, vol. 58, no. 11, pp. 1737–1747. doi: 10.1134/S0965542518110040
11. Il’in A.M. Matching of asymptotic expansions of solutions of boundary value problems. Providence: American Mathematical Society, 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.
12. Vishik M.I., Lyusternik L.A. A regular degeneration and boundary layer for linear differential equations with a small parameter. Uspekhi Mat. Nauk, 1957, vol. 12, no. 5, pp. 3–122 (in Russian).
13. 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. Asymptotics of a solution to a problem of optimal boundary control with two small cosubordinate parameters, Trudy Instituta Matematiki i Mekhaniki URO RAN, 2021, vol. 27, no. 2, pp. 108–119.
