Exact internal controllability for the semilinear heat equation (Q1363560)

The paper studies the exact controllability for the semilinear heat equation in a bounded domain \(\Omega\subset \mathbb{R}^n\): \[ y_t- \Delta y+ f(t,y)=h \quad \text{in } \Omega\times (0,T), \] \[ y(\cdot,0) =y^0 \text{ in } \Omega, \quad y|_{\partial \Omega \times (0,T)}= 0. \] Here, \(f\) is globally Lipschitz continuous in \(y\), and \(h\) is a control function. The problem is solved by introducing a nonlinear mapping \(F\). The authors first consider the problem: \[ u_t+ \Delta u=0, \quad u(\cdot,T) =u^T, \quad u |_{\partial \Omega \times (0, T)} =0, \] and then \[ y_t- \Delta y+f(t,y) =u-\Delta u, \quad y(\cdot,0) =y^0, \quad y |_{\partial \Omega \times (0,T)} =0. \] Set \(F(y^0,u^T) =y(\cdot,T)\). Via the property that \(F\) is Lipschitz continuous in \((y^0,u^T)\) and strongly monotone in \(u^T\), the Browder-Minty surjective theorem is applied to show that \(F\) is an onto-mapping. This implies the existence of a suitable \(T_0>0\) with the following property: Given \(y^0\), \(z^0\in L^2(\Omega)\) and \(0<T\leq T_0\), there is a control \(h=u- \Delta u\in L^2 (0,T;\;H^{-1} (\Omega))\) such that \(y(\cdot,T) =z^0\).