Carleman estimate for a parabolic equation in a Sobolev space of negative order and its applications. (English) Zbl 0977.93041

Chen, Goong (ed.) et al., Control of nonlinear distributed parameter systems. Partly proceedings of the conference advances in control of nonlinear distributed parameter systems, Texas A & M Univ., College Station, TX, USA. Dedicated to Prof. David L. Russell on the occasion of his 60th birthday. New York, NY: Marcel Dekker. Lect. Notes Pure Appl. Math. 218, 113-137 (2001).
The authors obtain a Carleman estimate for a second order parabolic equation, when the coefficients are not bounded and the right hand side is taken in \(L^2(0,T;H^{-1}(\Omega))\). Then, some applications are presented:
1) the global exact null-controllability of a semilinear parabolic equation, with a nonlinear term containing first order derivatives,
2) conditional stability in the continuation of the solution and
3) the inverse problem of determining a source term belonging to \(H^{-\ell}(\Omega)\) (with \(0 \leq \ell < 1\)) in a parabolic linear equation.
No proofs are provided for the Carleman estimate and the exact controllability result.
For the entire collection see [Zbl 0959.00047].


93C20 Control/observation systems governed by partial differential equations
93B05 Controllability
35K20 Initial-boundary value problems for second-order parabolic equations