Receding horizon control for the stabilization of the wave equation (Q1678206)

For stabilizing the wave equation, the equation is inserted into an optimal feedback control problem with an infinite horizon, a quadratic objective function, some boundary conditions and having a control input \(u=u(t)\), \(0 \leq t \leq + \infty\), acting with a control input operator \(B\) on the right hand side of the wave equation \(\ddot y - \Delta y = Bu\) (distributed control) or acting on parts of the boundary of the problem (Dirichlet, Neumann control, resp.). The given problems can be converted then into first order linear-quadratic control problems in a certain Hilbert space. An approximate optimal feedback control is constructed by the open-loop feedback control method which is also used in model predictive control/receding horizon control: Measuring the state \(z(t)\) of the problem at certain sampling time points \(t_k = \Delta k\), \(k=0,1,\dots\), optimal open-loop controls are determined at each finite time interval \([t_k,t_k + T]\) with a given time horizon \(T\). Sufficient conditions are provided for the suboptimality and exponential stability of the proposed approximate optimal feedback control procedure. Moreover, numerical simulations are given.