Carleman estimates and controllability results for the one-dimensional heat equation with BV coefficients
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.
scientific article; zbMATH DE number 5129693