Approximate controllability of the wave equation with mixed boundary conditions

Resumo

We consider initial boundary-value problem for acoustic equation in the time space cylinder Ω × (0; 2T) with unknown variable speed of sound, zero initial data, and mixed boundary conditions. We assume that (Neumann) controls are located at some part Σ Ω [0; T]; Σ ⊂ ????Ω of the lateral surface of the cylinder Ω × (0; T). The domain of observation is Σ × [0; 2T]; and the pressure on another part (????ΩnΣ) × [0; 2T]) is assumed to be zero for any control. We prove the approximate boundary controllability for functions from the subspace V ⊂ H¹(Ω) whose traces have vanished on Σ provided that the observation time is 2T more than two acoustic radii of the domain Ω. We give an explicit procedure for solving Boundary Control Problem (BCP) for smooth harmonic functions from V (i.e., we are looking for a boundary control f which generates a wave u^f such that u^f (., T) approximates any prescribed harmonic function from V ). Moreover, using the Friedrichs–Poincaré inequality, we obtain a conditional estimate for this BCP. Note that, for solving BCP for these harmonic functions, we do not need the knowledge of the speed of sound.