Abstract :
We consider the zero controllability problem for the distributed parameter
system governed by the damped wave equation
1 d du
ut tq2d x.uty p x. qq x.usF x, t., / r x. dx dx
where xgw0, ax, a-` with the boundary conditions
uxqkut. 0. s0, uxqhut. a. sw t., h, kgCj `4.
We consider two cases: a. F x, t.sg x.f t. and w t. is not identically zero;
b. w t.s0 and F x, t.s mjs1g j. x.fj t.. The functions f t., fj t., and w t. are
the distributed and boundary controls respectively. The problem with a nontrivial
function f t. and w t.s0 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 intervalw0, ax.
In the present paper, we investigate the conditions on the force profile functions
g x.,g j. x., js1, 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 w 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.,
js1, 2, . . . , m, under which the control time is equal to T1rm. 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. Q
Keywords :
nonharmonicexponential basis , Damping , boundary and distributed controls , exactcontrollability. , Nonselfadjoint operators , Generalized eigenvectors , Riesz basis
