# Mathematical Control Theory

Mathematical control theory is the area of applied mathematics dealing with the analysis and synthesis of control systems. To control a system means to influence its behaviour so as to achieve a desired goal such as stability, tracking, disturbance rejection or optimality with respect to some performance criterion.

The control theory group at Bath specialises in infinite-dimensional, nonlinear and sampled-data systems. The group organises a weekly seminar series and regularly hosts vistors. For more information, please visit our Control group web page.

Our research in mathematical control theory can be arranged under a number of broad headings, including:

- Absolute stability and input-to-state stability
- Sampled-data feedback systems
- Model reduction

Research on these topics has strong links with other areas of mathematics. Closely related areas include:

Further details of specific research areas are given below.

### Absolute stability and input-to-state stability

Research staff: Hartmut Logemann, Gene Ryan

The concept of absolute stability (going back to the late 1940s) permeates much of the classical and modern control literature and is applicable in the context of a canonical feedback scheme with a linear system in the forward path and a static nonlinearity in the feedback path, so-called Lure systems. The well-known circle and Popov criteria are of fundamental importance in absolute stability theory. The concept of input-to-state stability (ISS) is more recent and relates to stability properties of nonlinear systems with inputs. Since its inception in the 1980s, 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:

- input-to-state stability properties of Lure systems
- absolute stability and ISS in an infinite-dimensional context which allows applications to PDEs and FDEs
- absolute stability and ISS in the context of differential inclusions which allows applications to systems with discontinuous nonlinearities (such as, for example, Coulomb friction and quantization)
- absolute stability and ISS results for feedback systems with hysteresis nonlinearities
- the relevance of absolute stability theory and ISS in low-gain integral control
- application of discrete-time absolute stability and ISS results in the stability analysis of numerical methods for differential equations
- application of discrete-time absolute stability and ISS results in the context of models for biological invasions

### Sampled-data feedback systems

Research staff: Hartmut Logemann, Gene Ryan

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 have focussed 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 standard sample-and-hold operations)
- dynamic finite-dimensional sampled-data stabilization of linear infinite-dimensional systems with finite-dimensional unstable dynamics

### Model reduction

Research staff: Mark Opmeer

It is often desirable to replace an accurate but complex model for a physical 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 various types of balancing methods where the original complex model is assumed to be given by a partial differential equation. Analysis of error-bounds is of particular interest.