The cost of controlling weakly degenerate parabolic equations by boundary controls

Piermarco Cannarsa; Patrick Martinez; Judith Vancostenoble

doi:10.3934/mcrf.2017006

Article Contents

2017, Volume 7, Issue 2: 171-211. Doi: 10.3934/mcrf.2017006

This issue Previous Article Next Article Regularity results for a time-optimal control problem in the space of probability measures

The cost of controlling weakly degenerate parabolic equations by boundary controls

1.
Dipartimento di Matematica, Universitá di Roma "Tor Vergata", Via della Ricerca Scientifica, 00133 Roma, Italy
2.
Institut de Mathématiques de Toulouse, UMR CNRS 5219, Université Paul Sabatier Toulouse Ⅲ, 118 route de Narbonne, 31 062 Toulouse Cedex 4, France

* Corresponding author: Piermarco Cannarsa
* Corresponding author: Piermarco Cannarsa

Received: April 30, 2015

Revised: June 30, 2016

Published: August 2017

This research was partly supported by the Institut Mathematique de Toulouse and Istituto Nazionale di Alta Matematica through funds provided by the national group GNAMPA and the GDRE CONEDP. Moreover, this work was completed while the first author was visiting the Institut Henri Poincaré and Institut des Hautes Études Scientifiques on a CARMIN senior position.

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Abstract

We consider the one-dimensional degenerate parabolic equation

$u_t - (x^α u_x)_x =0 \;\;x∈(0,1),\ t ∈ (0,T) ,$

controlled by a boundary force acting at the degeneracy point $ x=0$.

We study the reachable targets at some given time $T$ using $ H^1$ controls, studying the influence of the degeneracy parameter $ α ∈ [0,1)$. First we obtain precise upper and lower bounds for the null controllability cost, proving that the cost blows up rationnally as $ \mathit{\alpha } \to {{\rm{1}}^{\rm{ - }}}$ and exponentially fast when $ \mathit{T} \to {{\rm{0}}^{\rm{ + }}}$.
Next, thanks to the special structure of the eigenfunctions of the problem, we investigate and obtain (partial) results concerning the structure of the reachable states.
Our approach is based on the moment method developed by Fattorini and Russell [19,20]. To achieve our goals, we extend some of their general results concerning biorthogonal families, using complex analysis techniques developped by Seidman [48], Guichal [26], Tenenbaum-Tucsnak [49] and Lissy [35,36].

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Mathematics Subject Classification: 35K65, 93B05, 33C10, 30B10.

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