Data assimilation and null controllability of degenerate/singular parabolic problems

Abstract

In this article, we use the variational method in data assimilation to study numerically the null controllability of degenerate/singular parabolic problem $$\displaylines{ \partial _{t}\psi - \partial_{x}(x^\alpha\partial _{x}\psi(x)) -\frac{\lambda }{x^{\beta }}\psi=f,\quad (x,t)\in ]0,1[\times]0,T[,\cr \psi(x,0)=\psi_0, \quad \psi\big|_{x=0}=\psi\big|_{x=1}=0. }$$ To do this, we determine the source term f with the aim of obtaining $\psi(\cdot ,T)=0$, for all $\psi_0 \in L^2(]0,1[)$. This problem can be formulated in a least-squares framework, which leads to a non-convex minimization problem that is solved using a regularization approach. Also we present some numerical experiments.

Keywords

• Data assimilation
• null controllability
• regularization
• heat equation
• inverse problem
• degenerate equations
• optimization