Resumen
Let : H be a 1 function on a real Hilbert space H and let >0 be a positive (damping) parameter. For any control function : ++ which tends to zero as t+, we study the asymptotic behavior of the trajectories of the damped nonlinear oscillator We show that if (t) does not tend to zero too rapidly as t+, then the term (t) x (t) asymptotically acts as a Tikhonov regularization, which forces the trajectories to converge to a particular equilibrium. Indeed, in the main result of this paper, it is established that, when is convex and S=argmin , under the key assumption that is a "slow" control, i.e., +0 (t) dt=+, then each trajectory of the (HBFC) system strongly converges, as t+, to the element of minimal norm of the closed convex set S. As an application, we consider the damped wave equation with Neumann boundary condition