Broadly, my research is centered around the theory, analysis and control of discontinuous dynamical systems, in addition to the development of associated numerical methods. Here I consider the term discontinuous dynamical systems very broadly to mean one in which the state evolution may exhibit discontinuities at certain times, or one for which the "generator" of the system has a lower order of smoothness.
Impulsive functional differential equations
Retarded functional differential equations -- RFDE for short -- are dynamical systems in which the immediate forward evolution depends explicitly on its prior history. These contrast to ordinary differential equations, for which the instantaneous rate of change depends only on the current state. The explicit way that the dynamics depend on the past often involve evaluations of the solution at delayed arguments, but they could also involve an integral term involving the history of the solution (a "distributed" delay). These equations are a natural setting to model phenomena in which there is a natural time lag, such as the incubation period of an infectious disease, a finite signal propagation time due to distance between sender and receiver, or the reaction of a control system to a user input.
Some real-world systems involve events that occur relatively infrequently but on very small time scales. It can sometimes be beneficial to model such events as occurring instantaneously -- that is, in zero time. If the system at hand also involves time lags, the result of such an approximation is an impulsive retarded functional differential equation (impulsive RFDE).
A large part of my research is devoted to understanding how the dynamics of impulsive RFDE can change when system parameters are varied. This falls under the heading of bifurcation theory. In this regard, centre manifold theory is an indispensable tool, allowing a reduction of dimension from the infinite to the finite. My research goals include the development of rigorous numerical methods to:
At a more theoretical level, the interactions between delays and impulses can result in some surprising, non-classical bifurcation patterns. The analysis of these bifurcations is complicated by technical issues relating to the non-smoothness of the centre manifold in terms of the two relevant parameters: the time delay and the period of the impulses. One of my goals is to develop techniques to identify such bifurcations and to classify them.
Invariant manifold-guided stabilization
Linear matrix inequalities (LMI) are indispensable tools in control theory and signal processing. Indeed, many solutions to classical control problems can be framed in terms of the existence of a solution to a LMI. When passing from a finite-dimensional problem like an ordinary differential equation in Euclidean space to a delay differential equation, however, even the problem of stabilizing an equilibrium becomes much more technical because of the infinitely many independent directions the state can travel, growing or decaying over time. Many authors have attempted to circumvent this problem by "projecting" the solution history into a finite-dimensional space, performing the control synthesis there using LMI methods, then attempting to reconstruct an appropriate controller for the infinite-dimensional system. Still other methods pose a stabilizing controller of a very specific form and verify that under certain LMI conditions, there is a Lyapunov functional that guarantees stability.
A recent interest of mine has been in using invariant manifold data to guide control synthesis in stabilization problems for infinite-dimensional systems. The centre-unstable manifold of the given system can be used to derive a finite-dimensional dynamical system that faithfully captures the effect of introducing a small, perturbative controller on the "unstable part" of the system. The advantage of designing the controller with this method is that all of the well-behaved stable modes can be ignored, with the result being a controller that can be more conservative than one derived with a LMI applied to a lossy finite-dimensional projection. This method also avoids the need for a non-canonical projection onto a finite-dimensional system or the construction of a complicated Lyapunov functional.
I have developed this invariant manifold-guided stabilization method for delay differential equations with linear impulsive controllers. There is provable feasibility when the unstable subspace is empty, and a heuristic is available for systems with nontrivial unstable subspace. A general purpose MATLAB implementation is available on this website, but in its current state the method only works well if the unstable eigenvalues are close to the imaginary axis and there is fairly large spectral gap between these and the stable eigenvalues. It also only supports a single delay. Further development will see: