Carleman Inequalities for Parabolic Equations in Sobolev Spaces of Negative Order and Exact Controllability for Semilinear Parabolic Equations

Abstract

We prove Carleman inequalities for a second order parabolic equation when the coefficients are not bounded and norms of right hand sides are taken in the Sobolev space L²(0, T;W₂^−{l}(Ω)), {l}∈[0, 1]. Our Carleman inequality yields the unique continuation for L²-solutions. We further apply these inequalities to the global exact zero controllability of a semilinear parabolic equation whose semilinear term also contains derivatives of first order of solutions and is of sub-linear growth at the infinity.