Some new results related to the null controllability of the 1-d heat equation
Séminaire Équations aux dérivées partielles (Polytechnique), (1997-1998), Talk no. 8, 22 p.

We address three null controllability problems related to the 1-d heat equation. First we show that the 1-d heat equation with a rapidly oscillating density is uniformly null controllable as the period of the density tends to zero. We also prove that the same result holds for the finite-difference semi-discretization in space of the constant coefficient heat equation as the step size tends to zero. Finally, we prove that the null controllability of the constant coefficient heat equation can be obtained as limit of null controllability properties for singularly perturbed dissipative wave equations. The proofs combine results on sums of real exponentials, Carleman’s inequalities for heat equations and sideways energy estimates for wave equations.

Nous étudions trois problèmes liés à la contrôlabilité à zéro de l’équation de la chaleur 1-d. Tout d’abord on montre que l’équation de la chaleur à dénsité périodique rapidement oscillante est uniformement contrôlable à zéro lorsque la période tend vers zéro. On démontre ensuite que les semi-discretisations en espace par différences finies de l’équation de la chaleur à coefficients constants sont uniformement contrôlables lorsque le pas de la discretization tend vers zéro. On montre aussi que la contrôlabilité à zéro de l’équation de la chaleur 1-d peut être obtenue comme une perturbation singulière des propriétés de contrôle à zéro d’équations d’ondes dissipatives.

@article{SEDP_1997-1998____A8_0,
     author = {L\'opez, Antonio and Zuazua, Enrique},
     title = {Some new results related to the null controllability of the $1-d$ heat equation},
     journal = {S\'eminaire \'Equations aux d\'eriv\'ees partielles (Polytechnique)},
     publisher = {Centre de math\'ematiques Laurent Schwartz, \'Ecole polytechnique},
     year = {1997-1998},
     note = {talk:8},
     mrnumber = {1660521},
     zbl = {1059.93058},
     language = {en},
     url = {http://www.numdam.org/item/SEDP_1997-1998____A8_0}
}
López, Antonio; Zuazua, Enrique. Some new results related to the null controllability of the $1-d$ heat equation. Séminaire Équations aux dérivées partielles (Polytechnique),  (1997-1998), Talk no. 8, 22 p. http://www.numdam.org/item/SEDP_1997-1998____A8_0/

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