Boundary and Distributed Controllability of the Damped Wave Equation: Reduction of Control Time

AbstractWe consider the zero controllability problem for the distributed parameter system governed by the damped wave equationutt+2dxut−1ρ(x)ddxp(x)dudx+qxu=Fx,t,where x∈[0,a], a<∞ with the boundary conditionsux+kut0=0,ux+huta=ϕt,h,k∈C∪∞.We consider two cases: (a) F(x,t)=g(x)f(t) and ϕ(t) is not identically zero; (b) ϕ(t)=0 and F(x,t)=∑mj=1g(j)(x)fj(t). The functions f(t), fj(t), and ϕ(t) are the distributed and boundary controls respectively. The problem with a nontrivial function f(t) and ϕ(t)=0 was studied in our recent (joint) work using the spectral decomposition method. The approach was based on the recent results of one of the authors on the spectral analysis of the nonselfadjoint dynamics generator of the system. It was shown that the system is controllable in a time T1 which is equal to twice the time it takes for a wave to travel between the ends of the interval [0,a]. In the present paper, we investigate the conditions on the force profile functions g(x),g(j)(x), j=1,2,…,m, under which the control time can be reduced. Namely, in case (a), we give the sufficient conditions on g(x) under which the combination of two controls f(t) and ϕ(t) allows us to reduce the control time twice. In case (b), we give the sufficient conditions on the force profile functions g(j)(x), j=1,2,…,m, under which the control time is equal to T1/m. In the proofs of these results, we use the generalization of the approach suggested by Russell for the boundary controllability of the undamped wave equations