Impulse controllability for degenerate singular parabolic equations via logarithmic convexity method

Abstract

Abstract In this paper, we study null approximate controllability of degenerate singular parabolic equations under the action of an impulsive control. To this aim, we prove an observation estimate at one point in time for the problems associated to the operators: $$ \begin{align*}& u_{t} -(x^{\alpha} u_{x})_{x} - \dfrac{\mu}{x^{\beta}} u = 0, \qquad x \in \left(0, 1\right), \end{align*} $$ where the parameters $\alpha \geq 0$, $\beta , \mu \in \mathbb{R}$ satisfy suitable assumptions. The method of proof combines both the logarithmic convexity and the Carleman commutator.