Abstract :
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 be a function on a real Hilbert space H, let γ>0 be a positive (damping) parameter and let 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 We established that, when Φ is convex and , under the key assumption that is a “slow” control, i.e., , 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 is sharp, indeed, when , 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 becomes (or behave) non L1?”Precisely, take a control term L1, let x be the solution of the corresponding (HBFC) system, take n to be a non increasing truncation of ( n(t)= (t) for t [0,n]), let x be the solution of the corresponding (HBFC) system. We show that 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.
