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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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