This field is centred on the subject of differential equations, which provide the basis for mathematical models in many fields including the physical sciences, engineering, the life sciences and finance. This subject is broad, ranging from the traditional ordinary and partial differential equations, to the more modern delay and stochastic differential equations. The two aspects of the subject that we emphasize are control theory and dynamical systems.
Firstly, control theory refers to the process of influencing the behaviour of a physical or biological system to achieve a desired goal, primarily through the use of feedback. The governing equations of the system in question are differential equations of various types. Secondly, the theory of dynamical systems deals with the qualitative analysis of solutions of differential equations on the one hand and difference equations on the other hand. The latter comprises the subfield of discrete dynamical systems, which has applications in diverse areas, for example biology and signal processing. A recent development is the notion of hybrid dynamical system, which allows the interaction of discrete events and continuous dynamics, thereby providing a natural framework for mathematical modeling of complex reactive systems or intelligent systems, in which physical processes interact with man-made automated environments.
The regular faculty whose primary research area is Control and Dynamical Systems are:
- S.A. Campbell (stability and bifurcation analysis of delay differential equations, mechanical systems with time delayed feedback)
- R. Guglielmi, (control of finite and infinite dimensional systems, optimization with PDE-constraints, model predictive control, stability of dynamical systems and feedback controls, applications to finance and natural sciences)
- J. Liu (hybrid systems and control, formal methods for control design, optimization and learning, applications in robotics, data science, and AI)
- X.Z. Liu (stability problems for nonlinear systems, numerical and control algorithms for hybrid dynamical systems, secure communication systems)
- K.A. Morris (control and estimation of systems governed by partial differential equations, including machine learning and other data-driven approaches to synthesis)
- N. Sri Namachchivaya (dynamical systems, stochastic stability and bifurcation analysis, nonlinear filtering, data assimilation, stochastic control and estimation)
- E.R. Vrscay (iterated function systems and applications, fractal-based analysis, mathematical imaging)
Cross-appointed faculty:
- A. Jagannath (dynamics in random landscapes, mathematics of data science, high-dimensional non-convex optimization, probability and stochastic analysis)
Emeritus faculty:
- D. Siegel (elliptic boundary value problems for PDEs, capillary surfaces, chemical kinetics of spatially inhomogeneous systems)
There are numerous links between Control and Dynamical Systems and the other research fields in the department. For example,
- Numerical methods for PDEs (Krivodonova, De Sterck, Morris)
- Members of the Environmental and Geophysical Fluids group study nonlinear wave equations (partial differential equations) in various contexts
- Ordinary differential equations, partial differential equations and dynamical systems form the basis for a variety of models in fluid dynamics and climate research (Lamb, Poulin, Stastna, Waite)
- Mathematical models in medicine and biology are based on various types of differential equations
- Mathematical models in image processing, e.g. self-similarity, resolution enhancement (Vrscay)
- Development of numerical algorithms for hybrid dynamical systems (Liu) and for capillary surfaces (collaboration between De Sterck and Siegel)
- Controller design links with Scientific Computing (J. Liu, Morris),
- Control theory links with systems biology (Ingalls) and quantum information processing (quantum control, Emerson and Morris)
- Deterministic trajectories in simple quantum mechanical systems (Vrscay)
Research groups: