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

Abstract

Let Φ:H→R be a C1 function on a real Hilbert space H and let γ>0 be a positive (damping) parameter. For any control function ε:R+→R+ 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))+ε(t)x(t)=0. We show that if ε(t) does not tend to zero too rapidly as t→+∞, then the term ε(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=argminΦ≠∅, under the key assumption that ε is a “slow” control, i.e., ∫+∞0ε(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]