Inertial Gradient-Like Dynamical System Controlled by a Stabilizing Term

Abstract

Let H be a real Hilbert space and let $$\Phi :H \to \mathbb{R}$$ be a $$\mathcal{C}^1$$ function that we wish to minimize. For any potential $$U:H \to \mathbb{R}$$ and any control function $$\varepsilon :\mathbb{R}_ + \to \mathbb{R}_ +$$ which tends to zero as t→+∞, we study the asymptotic behavior of the trajectories of the following dissipative system: $$({\text{S) }}\ddot x(t) + \gamma \dot x(t) + \triangledown \Phi (x(t)) + \varepsilon (t)\triangledown U(x(t)) = 0,{\text{ }}\gamma >{\text{0}}{\text{.}}$$ The (S) system can be viewed as a classical heavy ball with friction equation (Refs. 1–2) plus the control term ε(t)∇U(x(t)). If Φ is convex and ε(t) tends to zero fast enough, each trajectory of (S) converges weakly to some element of argmin Φ. This is a generalization of the Alvarez theorem (Ref. 1). On the other hand, assuming that ε is a slow control and that Φ and U are convex, the (S) trajectories tend to minimize U over argmin Φ when t→+∞. This asymptotic selection property generalizes a result due to Attouch and Czarnecki (Ref. 3) in the case where U(x)=|x|2/2. A large part of our results are stated for the following wider class of systems: $$({\text{GS) }}\ddot x(t) + \gamma \dot x(t) + \triangledown _x \Psi (t,x(t)) = 0,$$ where $$\Psi :\mathbb{R}_ + \times H \to \mathbb{R}$$ is a C 1 function.