Domain Decomposition in Optimal Final Value Control of Dissipative Wave Equations

Abstract

This chapter is concerned with domain decomposition methods for the computation of the solution of the optimality system associated with final value optimal control of second-order hyperbolic equations in which the control action enters through a dissipative boundary condition. More specifically, we consider optimal final value control of solutions of hyperbolic systems of the form

$$ \begin{gathered} \frac{{\partial ^2 w}} {{\partial t^2 }} - \nabla \cdot (A\nabla w) + cw = F in Q: = \Omega \times (0,T) \hfill \\ w = 0 on \Sigma ^D : = \Gamma ^D \times (0,T) \hfill \\ \frac{{\partial w}} {{\partial \nu _A }} + \alpha \frac{{\partial w}} {{\partial t}} = f on \Sigma ^N : = \Gamma ^N \times (0,T) \hfill \\ w(0) = w_0 , \frac{{\partial w}} {{\partial t}}(0) = v_0 in \Omega \hfill \\ \end{gathered} $$

((6.1.1.1))