The remarkable effectiveness of time-dependent damping terms for second order evolution equations

We consider a second order linear evolution equation with a dissipative term multiplied by a time-dependent coefficient. Our aim is to design the coefficient in such a way that all solutions decay in time as fast as possible. We discover that constant coefficients do not achieve the goal, as well as time-dependent coefficients that are too big. On the contrary, pulsating coefficients which alternate big and small values in a suitable way prove to be more effective. Our theory applies to ordinary differential equations, systems of ordinary differential equations, and partial differential equations of hyperbolic type.Comment: 32 pages, 5 figure

Stability of solutions to nonlinear wave equations with switching time-delay

In this paper we study well-posedness and asymptotic stability for a class of nonlinear second-order evolution equations with intermittent delay damping. More precisely, a delay feedback and an undelayed one act alternately in time. We show that, under suitable conditions on the feedback operators, asymptotic stability results are available. Concrete examples included in our setting are illustrated. We give also stability results for an abstract model with alternate positive-negative damping, without delay

Resonance effects for linear wave equations with scale invariant oscillating damping

We consider an abstract linear wave equation with a time-dependent dissipation that decays at infinity with the so-called scale invariant rate, which represents the critical case. We do not assume that the coefficient of the dissipation term is smooth, and we investigate the effect of its oscillations on the decay rate of solutions. We prove a decay estimate that holds true regardless of the oscillations. Then we show that oscillations that are too fast have no effect on the decay rate, while oscillations that are in resonance with one of the frequencies of the elastic part can alter the decay rate. In the proof we first reduce ourselves to estimating the decay of solutions to a family of ordinary differential equations, then by using polar coordinates we obtain explicit formulae for the energy decay of these solutions, so that in the end the problem is reduced to the analysis of the asymptotic behavior of suitable oscillating integrals.Comment: 27 pages, 1 table. In this version we modified the tile and we added Remark 2.7 following the feedback by some colleague

FAST CONVEX OPTIMIZATION VIA A THIRD-ORDER IN TIME EVOLUTION EQUATION

In a Hilbert space H, we develop fast convex optimization methods, which are based on a third order in time evolution system. The function to minimize f : H → R is convex, continuously differentiable, with argmin f = ∅, and enters the dynamic via its gradient. On the basis of Lyapunov's analysis and temporal scaling techniques, we show a convergence rate of the values of the order 1/t 3 , and obtain the convergence of the trajectories towards optimal solutions. When f is strongly convex, an exponential rate of convergence is obtained. We complete the study of the continuous dynamic by introducing a damping term induced by the Hessian of f. This allows the oscillations to be controlled and attenuated. Then, we analyze the convergence of the proximal-based algorithms obtained by temporal discretization of this system, and obtain similar convergence rates. The algorithmic results are valid for a general convex, lower semicontinuous, and proper function f : H → R ∪ {+∞}