A.R. Danilin. Asymptotics of a solution to a problem of optimal boundary control with two small cosubordinate parameters. II ... P. 108-119

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

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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.