Asymptotic Control and Stabilization of Nonlinear Oscillators with Non-isolated Equilibria

Let Φ: H→ be a 1 function on a real Hilbert space H and let γ>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(HBFC) x(t)+γx(t)+ Φ(x(t))+ (t) x(t)=0. 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 Φ is convex and S=argmin Φ≠ ︀, 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[formula]