Numerical estimations of the cost of boundary controls for the equation $y_t-\varepsilon y_{xx}+M y_x=0$ with respect to $\varepsilon$

Arnaud Münch (1)

1 LMBP - Laboratoire de Mathématiques Blaise Pascal

We numerically examine the cost of the null boundary control for the transport diffusion equation $y_t-\varepsilon y_{xx} + M y_x=0$, $x\in (0,L)$, $t\in (0,T)$ with respect to the positive parameter $\varepsilon$. It is known that this cost is uniformly bounded with respect to $\varepsilon$ if $T \geq T_M$ with $T_M\in [1, 2\sqrt{3}]L/M$ if $M>0$ and if $T_M\in [2\sqrt{2},2(1+\sqrt{3})]L/\vert M\vert$ if $M<0$. We propose a method to approximate the underlying observability constant and then conjecture, through numerical computations, the minimal time of controllability $T_M$ leading to a uniformly bounded cost. Several experiments for $M\in \{-1,1\}$ are performed and discussed.

Mots clés: Singular controllability; Lagrangian variational formulation; Numerical approxima- tion

Domaines: Optimisation et contrôle [math.OC]

Arnaud Münch. Numerical estimations of the cost of boundary controls for the equation $y_t-\varepsilon y_{xx}+M y_x=0$ with respect to $\varepsilon$. 2017. ⟨hal-01496856v2⟩