Numerical null controllability of semi-linear 1-D heat equations: fixed point, least squares and Newton methods
This paper deals with the analysis and the implementation of numerical methods allowing to solve the null controllability problem for the following semi-linear one-dimensional heat equation: \[ y_t-(D(x)y_x)_x+f(y)=v1_\omega,\,(x,t)\in (0,1)\times (0,T), \] \[ y(x,t)=0,\,(x,t)\in \{0,1\}\times (0,T); \quad y(x,0)=y_0(x),\,x\in (0,T), \] where \(T>0,\) \(\omega \subset\subset (0,1)\) is a non empty open interval, \(v\in L^\infty (\omega\times (0,T))\) is the control, \(y\) is the associated state and \(1_\omega\) is the characteristic function of the control set. The function \(D\) belongs to \(C^1([0,1])\) with \(D(x)\geq D_0>0\) and \(y_0\in L^2(0,1)\). The function \(f:\mathbb R\to\mathbb R\) is supposed to be at least locally Lipschitz-continuous. The goal is achieved by means of standard fixed point formulation, gradient techniques and Newton-Raphson strategy. Some data for which least squares algorithm converge are exhibited.
scientific article; zbMATH DE number 6104981