On time-optimal boundary control of vibrating beams

Abstract

For the vibration of a homogeneous beam it is shown, for a certain class of initial states at the time t=O, that null-controllability at t=T is possible for every time T>O, if control is applied to one of the boundary conditions on the right-hand side which is of sufficiently high order.

The controls v are assumed to be in H¹(O, T) and to satisfy v(O)=v(T)=O. The last condition guarantees that the beam stays in rest for all t≥T, if v is continued by putting v(t)=O.

It is further shown that restricted null-controllability at t=T is possible by controls v with ‖v'‖‖v′‖_L2_(0,T) being bounded by some given constant M>O, if T is sufficiently large. This guarantees the existence of time-minimal controls v_M for which it can be shown that ‖v′_M‖_L2_(0,T(M))=M where T(M) is the minimum time such that restricted null-controllability at t=T is possible.