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

By Hédy Attouch, Marc-Olivier Czarnecki. 2002

145 citations recorded by the source corpus.

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 e: 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))+e(t)x(t)=0. We show that if e(t) does not tend to zero too rapidly as t→+∞, then the term e(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 e is a “slow” control, i.e., ∫+∞0 e(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]

Subfields: Computational Theory and Mathematics, Control and Systems Engineering, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/8b4b4942b08fbad2bd57059da5899a7eb3fa7ddb

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Citations in the corpus

ASYMPTOTIC BEHAVIOR OF SECOND-ORDER DISSIPATIVE EVOLUTION EQUATIONS COMBINING POTENTIAL WITH NON-POTENTIAL EFFECTS ∗ (2009, background): “1, we recover various results which have been devoted to the Tikhonov dynamics, see [10,24] and the references therein.”
An inertial forward–backward splitting method for solving inclusion problems in Hilbert spaces (2018, unknown): “For second-order gradient systems involving inertial features, the introduction of a vanishing Tikhonov regularization was first considered by Attouch–Czarnecki [5] for the heavy ball with friction method.”
Asymptotic behavior of coupled dynamical systems with multiscale aspects (2009, unknown): “In [6] and [17] the authors show similar properties concerning the second order system (HBFC) ẍ(t) + γẋ(t) + ∇Ψ(x(t)) + ε(t)x(t) = 0 with γ > 0.”
A strong convergence result involving an inertial forward–backward algorithm for monotone inclusions (2017, background): “Let us recall that the inertial term is based upon a discrete version of a second order dissipative dynamical system [1,2] and can be regarded as a procedure of speeding up the convergence properties (see, e.”
On the long time behavior of second order differential equations with asymptotically small dissipation (2007, methodology): “Such a condition has already been pointed out for the steepest descent method combined with Tikhonov viscosity-regularization in convex minimization [3] as well as for the stabilization of nonlinear oscillators [4, 7].”