Asymptotic control and stabilization of nonlinear oscillators with non isolated equilibria, a note: from L1 to non L1

In [H. Attouch, M.-O. Czarnecki, Asymptotic control and stabilization of non-linear oscillators with non isolated equilibria, J. Differential Equations, 179 (2002) 278-310], we exhibited a sharp condition ensuring the efficiency of the Tikhonov-like control term in the (HBFC) system. Precisely, let Φ:H→R be a C1 function on a real Hilbert space H, let γ>0 be a positive (damping) parameter and let ɛ:R+→R+ be a control function which decreases to zero as t→+∞. In order to select particular equilibria in the important case where Φ has non isolated equilibria, we introduced in [H. Attouch, M.-O. Czarnecki, J. Differential Equations, 179 (2002) 278-310] the following damped nonlinear oscillator and studied its asymptotic behavior(HBFC)x¨(t)+γx˙(t)+∇Φ(x(t))+ɛ(t)x(t)=0.We established that, when Φ is convex and S=argminΦ≠∅, under the key assumption that ɛ is a “slow” control, i.e., ∫0+∞ɛ(t)dt=+∞, then each trajectory of the (HBFC) system strongly converges, when t→+∞, to the element of minimal norm of the closed convex set S. The condition on the control term ɛ is sharp, indeed, when ∫0+∞ɛ(t)dt<+∞, the trajectory weakly converges but it may not strongly converge and we have no information a priori on the weak limit. In this note, we give an answer to the following question: “When does an L1 control term ɛ becomes (or behave) non L1?”Precisely, take a control term ɛ∉L1, let x be the solution of the corresponding (HBFC) system, take ɛn to be a non increasing truncation of ɛ (ɛn(t)=ɛ(t) for t∈[0,n]), let x be the solution of the corresponding (HBFC) system. We show thatlimn→+∞∥xn-x∥∞=0.In particular, the weak limits of the trajectories xn strongly converge, when n→+∞, to the (strong) limit of the trajectory x. In other words, there is no loss of the information gained by the “slow behavior” for t⩽n.