Section: New Results

Nonlinear systems with delay

1) In [45] , we developed a new technique for stability analysis for nonlinear dynamical systems with delays and possible discontinuities. In contrast with Lyapunov based approaches, the trajectory based approach we proposed involves verifying certain inequalities along solutions of auxiliary systems. It applies to a wide range of systems, notably time-varying systems with time-varying delay, ODE coupled with difference equations, and networked control systems with delay. It relies on the input-to-state stability notion, and yields input-to-state stability with respect to uncertainty.

2) In [39] , to address various types of delays including the neutral-type arising in dynamical networks, we dealt with coupled delay differential and continuous-time difference equations and proposed stability and robustness criteria. In these criteria, differential equation parts do not necessarily exhibit unbounded dissipation rate. Subsystems described by differential equations are not required to be input-to-state stable either. No assumptions on network topology are made. To handle such a general case, we construct explicit Lyapunov-type functionals. We established stability and robustness of the overall networks.

3) In [22] , [42] and [44] , stability results for several families of systems with delay are established. The key ingredient of these contributions is the use of comparison systems of a new type, the theory of the positive systems and linear Lyapunov functionals. We provided robustness of the stability with respect to multiplicative uncertainty in the vector fields. We allowed cases where the delay may be unknown, and where the vector fields defining the systems are not necessarily bounded. We illustrate our work using a chain of integrators and other examples.