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Carleman estimates and controllability results for the one-dimensional heat equation with BV coefficients. (English) Zbl 1128.35020
Global Carleman estimates for one-dimensional linear parabolic equations with a coefficient of bounded variations are derived. First, the author constructs limit weight functions by approaching the bounded variation coefficient \(c\) by piecewise constant coefficients \(c_{\varepsilon}\). Then he proves a Carleman estimate associated to \(\partial_t \pm \partial_x (c\partial_x)\) by proving that the constants in the Carleman estimate \(\partial_t \pm \partial_x (c_{\varepsilon}\partial_x)\) can be taken uniform with respect to the parameter \(\varepsilon\) and passing to the limit in each term of the estimate. In the conclusive part of the work, the author derives a Carleman estimate for a linear parabolic system with the right-hand side in \(L^2(0,T,H^{-1}(\Omega))\). This estimate is used for the analysis of the controllability of a semilinear system.

MSC:
35B37 PDE in connection with control problems (MSC2000)
93B05 Controllability
35K20 Initial-boundary value problems for second-order parabolic equations
35B45 A priori estimates in context of PDEs
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