## Mathematical Control Theory Group |

Research areas of the mathematical control theory group are:

- Control of Positive Systems including Applications to Mathematical Biology
- Infinite-Dimensional Control Systems
- Model Reduction
- Nonlinear Control Systems and Input-to-State Stability
- Sampled Data Feedback Systems
- Applications of Operator Theory in Systems and Control

Positive dynamical systems, or simply positive systems, are dynamical systems where the evolution map leaves a positive cone invariant, such as the nonnegative orthant of Euclidean space, equipped with the partial order of component wise inequality. The study of positive systems is motivated by numerous applications across a diverse range of scientific and engineering contexts where invariance of a positive cone capture the essential property that state-variables of positive systems must take nonnegative values to be meaningful. The breadth and depth of applications of positive and (related) monotone systems has generated significant interest in their control. Our research focusses on:

- stability of nonlinear positive dynamical systems – both forced and unforced;
- the relaxation of classical positive input control systems to the special case where negative controls are permitted, provided that nonnegativity of the state is not violated, and;
- applications of 1. and 2. to problems and societal challenges arising in biology and ecology, such as dynamic resource management (crop pests and pest pathogens, for example).

The control of infinite-dimensional, or so-called distributed-parameter, systems, such as dynamical systems specified by controlled and observed partial or delay differential equations is motivated by:

- their prevalence in applications, and;
- the mathematical challenges caused by infinite-dimensionality.

Of particular interest to our research group is the development of aspects of the other listed research areas (for example: model reduction, nonlinear control systems and input-to-state stability) in infinite-dimensional settings.

It is often desirable to replace an accurate but complex model for a physical, biological or engineering system by a perhaps slightly less accurate but simpler model. The process to extract the simpler model from the more complex one is called model reduction. Our research on model reduction mainly focuses on two cases:

- the case where the original complex model is given by partial differential equations;
- the case where the original model has a structure (such as positivity or passivity) which must be retained in the simpler model.

The concept of input-to-state stability (ISS) relates to stability properties of forced nonlinear systems (the forcing is frequently called input). Since its inception in 1989, this concept has generated a rich body of results, in particular, extending classical Lyapunov theory to systems with inputs. Our work in this area focuses on the following aspects:

- ISS and classical absolute stability theory;
- ISS in an infinite-dimensional context with applications to controlled PDEs and FDEs;
- ISS in the context of differential inclusions which allows applications to systems with discontinuous nonlinearities (such as Coulomb friction and quantization);
- ISS results for hysteretic feedback systems;
- ISS methods in the context of population dynamics.

A sampled-data feedback system consists of a continuous-time system, a discrete-time controller and certain continuous-to-discrete (sample) and discrete-to-continuous (hold) operations. Sampled-data control theory forms the mathematical foundation of digital control which addresses the problem of implementing control strategies using digital computers. Our activities focus on:

- sampled-data integral controllers for linear infinite-dimensional systems subject to input and output nonlinearities (possibly of hysteresis type);
- dynamic low-gain finite-dimensional sampled-data controllers for stable linear infinite-dimensional systems, achieving approximate asymptotic tracking and disturbance rejection;
- indirect sampled-data stabilization of linear infinite-dimensional systems (that is, sampled-data stabilization via discretization of a stabilizing continuous-time controller using suitable sample-and-hold operations);
- dynamic finite-dimensional sampled-data stabilization of linear infinite-dimensional systems with finite-dimensional unstable dynamics.

Many aspects of operator theory play a role in systems and control. Our work includes the application of and the further development of operator theory related to:

- semigroups of linear operators;
- Hankel operators;
- operator-valued holomorphic functions;
- optimization in Hilbert spaces;
- positive operators;
- operator Lyapunov and Riccati equations;
- dissipative operators in both definite and indefinite inner-product spaces.

Mathematical Control Theory Group, Department of Mathematics, University of Bath