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Global Carleman estimates for solutions of parabolic systems defined by transposition and some applications to controllability. (English) Zbl 1113.35037

The paper is concerned with application of Carleman estimates applied to exact controllability of the linear parabolic system \[ -\varphi_t - \triangle\varphi=f-\nabla F + \displaystyle{\sum_{i,j=1}^N\partial_{ij}H_ij-G_t \;,\;in\;Q=\Omega\times [0,T]\subset \mathbb R^N\times [0,T]} \]
\[ \varphi=0\;on\;\partial\Omega\;,\;\varphi(T)=\varphi^o\;in\;\Omega \] with \(f,G\in L^2(Q)\;,\;F\in L^2(Q)^N\;,\;G\in C^o([0,T];H^{-1}(\Omega))\;,\;H_{ij}\in L^2(Q)\), the derivatives being taken in the generalized sense. The same problem is then considered for the nonlinear parabolic system \[ y_t - \triangle y - \varepsilon\displaystyle{\sum_{i,j=1}^Ng_{ij}(\chi,t;y,\nabla y)\partial_{ij}y=\nu 1_{\omega} \;in\;Q} \]
\[ y = 0 \;on\;\Sigma=\partial\Omega\times[0,T]\;,\;y(0)=y^o\;on\;\Omega \] with \(\omega\) some small nonempty open subset of \(\Omega\).

MSC:

35B45 A priori estimates in context of PDEs
35K55 Nonlinear parabolic equations
35B37 PDE in connection with control problems (MSC2000)
35K50 Systems of parabolic equations, boundary value problems (MSC2000)
93B05 Controllability
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