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The cost of controlling weakly degenerate parabolic equations by boundary controls

  • * Corresponding author: Piermarco Cannarsa

    * Corresponding author: Piermarco Cannarsa 
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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  • 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].

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


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