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In this paper we prove an inverse inequality for the parabolic equation in a bounded domain Ω⊂ with Dirichlet boundary conditions. With the motivation of finding an estimate of g in terms on the trace of the solution in 𝒪×(0,T) for ℇ small, our approach consists in studying the convergence of the solutions of this equation to the solutions of some transport equation when ℇ→0, and then recover some inverse inequality from the properties of the last one. Under some conditions on the open sets ω, 𝒪 and the time T, we are able to prove that, in the particular case when g∈H01(ω) and it does not depend on time, we have: |g|L2(ω)≤C(|u|H1(0,T;L2(𝒪))+ℇ1/2|g|H1(ω)). On the other hand, we prove that this estimate implies a regional controllability result for the same equation but with a control acting in 𝒪×(0,T) through the right-hand side: for any fixed f∈L2(ω) , the L2-norm of the control needed to have |u(T)|ω-f|H-1(ω)≤γ remains bounded with respect to γ if ℇ≤Cγ2. [ABSTRACT FROM AUTHOR]

We consider the problem of the null controllability of a family of 1-D linear control parabolic equations depending on two parameters, namely the viscosity and the coefficient of the transport term. We study the dependence, with respect to these parameters and the time of controllability, of the norm of the optimal controls. In particular we give estimates on the optimal control as the viscosity tends to 0. [ABSTRACT FROM AUTHOR]