Let \(\Omega\subset\mathbb{R}^n\) be a bounded domain, \(\varepsilon\in(0,1)\) a small parameter and \(\xi=\{\xi_i\}_{i=1}^\infty\in \ell_2\) a fixed vector. Set \(Q=(0,\infty)\times\Omega,\) and consider the optimal control problem \[ \begin{cases} \frac{dy}{dt}=A^\varepsilon y(t)+u(t,x), & (t,x)\in Q,\\ y|_{\partial\Omega}=0, & \\ y|_{t=0}=y^\varepsilon_0, \end{cases} \]
\[ \inf \int_0^\infty\int_\Omega \big(y^2(t,x)+u^2(t,x)\big)dtdx, \]
\[ u\in U_\varepsilon=\left\{ v\in L^2(Q):\;\forall i\geq 1 \;\left|\int_\Omega v(t,x)X_i^\varepsilon(x)dx\right|\leq \xi_i\;\text{for a.a.}\;t>0\right\}. \] Here \(A^\varepsilon=\text{div}(a^\varepsilon\nabla),\) \(a^\varepsilon=a\left(\frac{x}{\varepsilon}\right),\) \(a\) is a measurable, symmetric, periodic and uniformly elliptic matrix, while \(\{X^\varepsilon_i\}\) and \(\{\lambda_i^\varepsilon\}\) are solutions of the spectral problem \[ \begin{cases} A^\varepsilon X^\varepsilon_i=-\lambda^\varepsilon_i X^\varepsilon_i,\\ X^\varepsilon_i|_{\partial\Omega}=0, \end{cases} \] \(\{X^\varepsilon_i\}\subset H^1_0(\Omega)\) is an orthonormal basis in \(L^2(\Omega),\) and \(0<\lambda^\varepsilon_1\leq \lambda^\varepsilon_2\leq\ldots\) with \(\lambda^\varepsilon_i\to\infty\) as \(i\to\infty.\)
In the note under review, the authors obtain a feedback formula for the distributed optimal control by justifying the form of the approximate optimal regulator \(u[y]\) of the above problem.
For the entire collection see [Zbl 1357.00036].