**Functional Differential Equations and Time Delay Systems**

A delay system is described by functional differential equations, and is an infinite dimensional system. We studied modelling and stability of physical systems with time delays. These delays may be constant, time-varying or more generally, dependent on the state of the system itself. One interest is in the well-posedness of mathematical models of systems with delays, without violating causality. Our recent research in this area focuses on control, observation and optimization for delay systems.

One application in physics is the analysis of post-Newtonian gravitation as an approximation to Einsteinâ€™s field equations for low speeds and small fields. A down to earth application involves improved position estimation in marine and dep space environments.

**Model Reduction and System Structure**

We investigated the topological structure of periodic and more general switched systems. For time varying systems we developed a methodology for model reduction and extend this to the realm of smooth nonlinear systems. As opposed to linear systems, it was found that viable nonlinear reduced models must be of a hybrid (switched system) nature.

An integral part is the investigation of causal models for systems with jumps in their state.

For interconnected linear systems we have developed new notions of partial reachability and observability and

Using the language of polynomial system theory, we develop criteria for these generalized concepts

**Optimal Control**

We are interested in optimal control for hybrid systems, and delay systems, and optimization for Maximum accuracy and minimum sensitivity. We are also looking into applications in epidemiology and ecology. In particular we investigate resource allocation problems in the population dynamics of bee colonies.

**Path Planning and Locomotion in Robotics**

Although this research is mainly of a theoretical nature, using methods from algebraic geometry, the application domains include robotics and biologically inspired control systems.