Broadly, my research interests are centered around the theory, analysis and control of nonsmooth dynamical systems, applications in the sciences, and rigorous numerics for infinite-dimensional dynamical systems. Here I consider the term nonsmooth 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. Secondary research areas include topics in functional differential equations and bifurcation theory in more general terms.
Nonsooth dynamical systems and rigorous numerics
Nonsmooth dynamical systems is a relatively new field whose development was motivated in part by its ubiquity in control engineering, but whose applications are now incredibly diverse. These systems spend most of their time evolving in a smooth manner, but under special circumstances their evolution can be interrupted in a discontinuous way. A simple example is a bouncing ball: it falls smoothly until it hits the ground, at which point its direction of travel is quickly reversed.
From the perspective of simulation, such systems are more difficult to study with high precision because their structure frequently requires the numerical solver to be interrupted to either reset the state or verify zero-crossing conditions. These difficulties are further compounded if the continuous dynamics are governed by an infinite-dimensional dynamical system.
Much of my current research concerns the development of rigorous numerics for nonsmooth dynamical systems. Rigorous numerics is often equated with computer assisted proofs. While proofs are the main draw of the rigorous numerics philosophy, a concrete advantage is that we usually get a numerical method for free after all of the work is done. If rigorous bounds on the error of the solution of the problem are required, these can be obtained by post-processing of the result of the computer-assisted proof.
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 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). They are a subclass of infinite-dimensional discontinuous dynamical systems.
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. Some of my long-term research goals are to make this reduction more computationally viable by developing 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.
Finally, the theory of impulsive functional differential equations is still in development, particularly for state-dependent delays and systems with impulses triggered by state-dependent relationships. One of my research goals is to study the Cauchy problem for these more general classes of systems and develop stability and invariant manifold theory for them, with a view toward bifurcations.
Infectious disease modeling
Nonsmooth dynamical systems ideas are very applicable to control, so it is natural that they have found use in infectious disease modeling. As for my own research, some recent project include the topics
Pandemic closing a reopening: In early 2020, the SARS-CoV-2 pandemic made it necessary for governments worldwide to impose lockdown conditions. This allowed us to collectively "flatten the curve", but was difficult for businesses and individuals. In spring and summer of that year, some amount of "reopening" allowed greater individual freedoms. This then caused further outbreaks and in the fall of 2020, many governments began lockdowns again. This "closing and reopening" can be modeled by relays. Here, the model involves a switching vector field, which is a type of discontinuous dynamical system. The discontinuity comes from lockdowns being imposed and lifted based on the progression of the pandemic, specifically a moving average of the number of active cases. The result is an implicitly-defined infinite-dimensional dynamical system. I completed the analysis of such a model, proving the existence of cycles with repeated closing and reopening events.
Pulse vaccination with finite immunity: Pulse vaccination is a vaccination strategy whereby a large proportion of the population is vaccinated in a short period of time. This event is repeated after a set amount of time. It was unclear how effective such strategies are when the vaccine or clearance of the infection confers immunity for only a set amount of time. I analyzed a model that incorporated both of these effects. I found that if not enough people get vaccinated then smaller outbreaks may become more common year-round. These smaller outbreaks have a temporally quasiperiodic structure, so while they are oscillatory it is more difficult to anticipate them.
Invariant manifold-guided control and 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 -- that is, only slow-unstable centre directions are present -- and a heuristic is available for systems with nontrivial unstable subspace. A general purpose MATLAB implementation is available on this website. In its current state the method 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 currently supports a single delay. Further development will see: