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.