CONTROLLABILITY OF THE WAVE-EQUATION WITH MOVING POINT CONTROL

This paper deals with approximate and exact controllability of the wave equation in finite time with interior point control acting along a curve specified in advance in the system's spatial domain. The structure of the control input is dual to the structure of the observations which describe the measurements of velocity and gradient of the solution of the dual system, obtained from the moving point sensor. A relevant formalization of such a control problem is discussed, based on transposition. For any given time-interval [0, T] the existence of the curves providing approximate controllability in H(D)-[n/2]-1 (OMEGA) x H(D)-[n/2]-2(OMEGA) (where n stands for the space dimension) is established with controls from L2(0, T; R(n+1)). The same curves ensure exact controllability in L2(OMEGA) x H-1 (OMEGA) if controls are allowed to be selected in [L(infinity)(0, T; R(n+1))]'. Required curves can be constructed to be continuous on [0, T).