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Singular optimal control for a transport-diffusion equation. (English) Zbl 1135.35017
The authors consider the following controlled linear transport-diffusion equations
\[ \begin{aligned} y_t-\varepsilon\Delta y+ M(x,t)\cdot\nabla y= u_1\mathbf{1}_\omega&\quad\text{in }\Omega\times (0,T),\\ y= 0&\quad\text{on }\partial\Omega,\\ y(x, 0)= y^0(x)&\quad\text{in }\Omega\end{aligned}\tag{1} \]
and
\[ \begin{aligned} y_t-\varepsilon\Delta y+ M(x, t)\cdot\nabla y= 0&\quad\text{in }\Omega\times (0,T),\\ y= u_2\theta(x)&\quad\text{on }\partial\Omega,\\ y(x,0)= y^0(x)&\quad\text{in }\Omega,\end{aligned}\tag{2} \]
where \((\varepsilon, T)\in (0,+\infty)^2\) and \(\Omega\) is a bounded domain in \(\mathbb{R}^N\). Here \(u_1\) and \(u_2\) stand for the control functions which act over the system (1) through the (nonempty) open subset \(\omega\subset\Omega\) and a (nonempty) part of the boundary in the case of problem (2). In the case of problem (2), \(\theta\geq 0\) is a smooth, not identically zero, cut-off function on the boundary.
In this paper the authors are interested in the cost of the null controllability of systems (1) and (2) as \(\varepsilon\to 0\). Firstly, they prove that, under suitable (and natural) hypotheses on \((T,M)\), the constant that quantifies the cost of the null controllability of (1) and (2) grows exponentially as \(\varepsilon\to 0\). In a second step the authors prove that both costs are exponentially small as \(\varepsilon\to 0\) whenever \(M\) satisfies a suitable geometric condition and the control time \(T\) is large enough.

MSC:
35B37 PDE in connection with control problems (MSC2000)
35K05 Heat equation
93B05 Controllability
35B25 Singular perturbations in context of PDEs
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References:
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