Null controllability of one-dimensional parabolic equations by the flatness approach (Q2789582)

The paper is concerned with linear one-dimensional parabolic equations with space dependent coefficients that are only measurable and that may be degenerate or singular. The authors prove the null controllability with one boundary control with the use of the flatness approach, which provides explicitly the control and the associated trajectory as series. Both the control and the trajectory have a Gevrey regularity in time related to the \(L^p\) class of the coefficient in front of \(u_t\). The basic result of the paper is presented in Theorem~1.1. The approach applies in particular to the (possibly degenerate or singular) heat equation \((a(x)u_x )_x - u_t =0\) with \(a(x)>0\) for a.e. \(x\in(0,1)\) and \(a+1/a\in L^1 (0,1)\), or to the heat equation with inverse square potential \(u_{xx} +(\mu/|x|^2 )u-u_t =0\) with \(\mu\geq 1/4\). The sufficient conditions of controllability of this equation are given in Theorem~1.3.