Optimal boundary control of the heat equation with target function at terminal time.
The paper considers the problem of minimizing the functional \[ I= \int_\Omega| y(T, x)- h(x)|^2\,dx+ \int^T_0 \int_{\partial\Omega}| u(t,x)|^2\,dS\,dt \] subject to \[ \begin{gathered} y_t=\Delta y\quad\text{in }(0,T)\times \Omega,\quad y(0,\cdot)= y_0(\cdot),\\ y(t, x)= u(t,x)\quad\text{on }(0,T)\times \partial\Omega,\end{gathered} \] where \(\Omega\subset\mathbb{R}^2\) is a Lipschitz domain with boundary \(\partial\Omega\), \(y_0\) and \(h\) are given functions. The authors reformulate the original problem as an abstract one, derive the optimality system and obtain the optimal feedback by using the formal solution of the corresponding Riccati equation.
scientific article; zbMATH DE number 1861049