Global Carleman estimates for solutions of parabolic systems defined by transposition and some applications to controllability

Abstract

Let Ω ⊂ R (N ≥ 1) be a bounded connected open set whose boundary ∂Ω is regular enough (e.g., ∂Ω ∈ C). Let ω ⊂ Ω be a (small) nonempty open subset and let T > 0. We will use the notation Q = Ω × (0, T) and Σ = ∂Ω × (0, T) and we will denote by n(x) the outward unit normal toΩ at the point x ∈ ∂Ω. In this paper we deal with the controllability properties of some nonlinear parabolic equations for which the nonlinear terms are related to time derivatives and/or second-order spatial derivatives. As usual,we will first strict ourselves to similar linear systems and then we will be concerned with the original nonlinear problems. For the controllability analysis of the linear systems, the main tool will be a new Carleman estimate that holds for very weak solutions, that is, for solutions that only belong to L(Q) = L(0, T ; L(Ω)), of appropriate linear parabolic systems. The sense we will give to these solutions comes from the formulation by transposition of the corresponding systems. More precisely, let us assume that φ ∈ H(Ω) and f, F, G, andH satisfy