Research Areas
- Hybrid Dynamical Systems
- Switched Systems
- Mixed-Integer Optimal Control
- Control of Partial Differential Equations
My research interests focus on the modeling, analysis and control of switched systems and hybrid dynamical systems with modes that are governed by partial differential equations. The motivation originates in the observation that system theoretical descriptions of many physical problems involve variables that are modeled as discrete. A gear in a car model or a signal in a traffic flow model are typical examples. A gene's rate of expression in a genetic regulatory network is a less obvious one. The dynamics of those discrete variables within a continuously evolving system then shows up as switching. In most cases, such models are an ad-hoc description of much more complex processes on finer time scales and thus they have a number of limitations, but their study can nevertheless provide valuable qualitative insight to the system's behavior on the coarse scale. These discrete-continuous dynamical systems are therefore an already much-studied area of interest in the lumped parameter (ordinary differential equation) context. In particular, the use of the discrete variable as a control mechanism is useful for many applications. For distributed parameter problems governed by partial differential equations, very little is known in this context. The analysis that I carry out concerns solution concepts, well-posedness theory and the study of qualitative behavior such as asymptotic stability on the discrete-continuous level of these multiscale dynamical systems. Control refers to problems where the switching is open-loop and subject to optimization, or closed-loop and determined by decision rules. Most of my contributions apply to problems involving systems of (semi-)linear hyperbolic equations which model, for instance, a variety of transport processes on networks and involve interaction of pumps, valves and other fast control elements located at the nodes. This includes systems with modes governed by conservation laws and balance laws. Recently, I'm investigating a more general theory concerning switched and hybrid dynamical systems with modes governed by abstract evolution processes on infinite dimensional spaces. Such a theory covers, in particular, multiscale problems with bio-chemical applications that can be modeled using switching (semi-)linear parabolic equations or problems in quantum mechanics with large uncertainties modeled as switching linear Schrödinger equations.