No abstract

We consider the wave equation with a time-varying delay term in the boundary condition in a bounded and smooth domain Ω ⊂ IR n . Under suitable assumptions, we prove exponential stability of the solution. These results are obtained by introducing suitable energies and suitable Lyapunov functionals. Such analysis is also extended to a nonlinear version of the model.

In this paper we consider a boundary stabilization problem for the wave equation with interior delay. We prove an exponential stability result under some Lions geometric condition. The proof of the main result is based on an identity with multipliers that allows to obtain a uniform decay estimate for a suitable Lyapunov functional. (2000): 35B05, 93D15, 93D20 Mathematics Subject Classification

We consider the celebrated Cucker-Smale model in finite dimension, modelling interacting collective dynamics and their possible evolution to consensus. The objective of this paper is to study the effect of time delays in the general model. By a Lyapunov functional approach, we provide convergence results to consensus for symmetric as well as nonsymmetric communication weights under some structural conditions.

We analyze the Cucker-Smale model under hierarchical leadership in presence of a time delay. By using a Lyapunov functional approach and some induction arguments we will prove convergence to consensus for every positive delay τ. We also prove a flocking estimate in the case of a free-will leader. These results seem to point out the advantage of a hierarchical structure in order to contrast time delay effects that frequently appear in real situations.