Optimal Final Value Boundary Control of Conservative Wave Equations

This chapter is concerned with the problem with of optimal final value control of the second-order hyperbolic system (8.1.1.1)$$ \begin{gathered} \frac{{\partial ^2 w}} {{\partial t^2 }} - \nabla \cdot (A\nabla w) + cw = F in \Omega \times (0,T) \hfill \\ w = 0 on \Gamma ^D \times (0,T) \hfill \\ \frac{{\partial w}} {{\partial \nu _A }} = f on \Gamma ^N \times (0,T) \hfill \\ w(0) = w_0 , \frac{{\partial w}} {{\partial t}}(0) = v_0 in \Omega . \hfill \\ \end{gathered} $$ where f ∈ L2(ГN × (0,T)) is the control input and F is a given distributed system input. We retain the notation of Section 6.2.1, as well as the assumptions on the coefficients A and c, the region Ω and its boundary components ΓD, ΓN delineated there. When f =0 andF =0, this system conserves energy: E(t) ≡ E(0), where the energy functional E(t) is given by $$ E(t) = \int_\Omega {\left[ {\left| {\frac{{\partial w}} {{\partial t}}(x,t)} \right|^2 + A\nabla w(x,t) \cdot \nabla w(x,t) + c\left| {w(x,t)} \right|^2 } \right]} dx. $$