Mark Opmeer

Senior Lecturer in Mathematics

Department of Mathematical Sciences
University of Bath
Claverton Down
Bath BA2 7AY

Office: 4 West 2.14

My research interests are I am especially interested in the intersection of the above research domains.

Research themes

My current research can be roughly grouped into a number of themes: Some of my earlier publications were on the following themes:

Infinite-dimensional Lur'e systems

Lur'e systems consist of a linear part in feedback with a static nonlinearity. Together with Chris Guiver and Harmut Logemann (Guiver, Logemann and Opmeer 2019) I've obtained a circle-criterion type condition for input-to-state stability for infinite-dimensional Lur'e systems. In connection with that result, we studied operator-valued positive real functions in (Guiver, Logemann and Opmeer 2017).

Model reduction

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.
My research on model reduction mainly focuses on the case where the original complex model is given by partial differential equations. Analysis of convergence and error-bounds is of particular interest.

Linear Quadratic optimal control

In a series of articles with Olof Staffans, the Linear Quadratic optimal control problem for infinite-dimensional systems is considered at a very high level of generality. The first three articles in the series dealt with discrete-time systems. The first article in the series is (Opmeer and Staffans 2008), where we treat the linear quadratic optimal control problem on the positive time axis, right factorizations and unbounded solutions of the control Riccati equation. In the second article in the series (Opmeer and Staffans 2010) we treat the linear quadratic optimal control problem on the negative time axis, left factorizations and unbounded solutions of the filter Riccati equation. The third article in the series (Opmeer and Staffans 2012) deals with the linear quadratic optimal control problem on the doubly infinite time axis and strongly coprime factorizations. There a coupling condition connecting the control and the filter Riccati equation shows up. The fourth article (Opmeer and Staffans 2014) contains the corresponding continuous-time results. The fifth article (Opmeer and Staffans 2019) considers the well-posed case.

Numerical methods for Lyapunov, Riccati and Lur'e equations

With Timo Reis (and additional collaborators) I've studied the ADI method for solving Lyapunov and Riccati equations. In (Opmeer, Reis and Wollner 2013) the ADI method for numerically solving Lyapunov equations is studied. We prove convergence and illustrate how adaptive finite elements can be used to reduce computational time. In (Massoudi, Opmeer and Reis 2016) we prove convergence of the generalization of ADI to Riccati equations and in (Massoudi, Opmeer and Reis 2017 ) we extend ADI to bounded real and positive real Lur'e equations and again prove convergence.

State space methods in the factorization approach to controller design

One of the main problems in control theory is to find a (robustly) stabilizing controller for a given system. It is often desirable to have state space formulas for such a stabilizing controller.
Such state space formulas can be obtained from the solution of the linear quadratic optimal control problem and involve the solutions of algebraic Riccati equations. As intermediate results state space formulas for Bezout factors and for the solution of the Nehari problem have to be obtained.
In my research into these problems the objective is to use minimal assumptions so as to cover as wide a class of (partial differential equation) systems as possible.

Distributional control systems

Usually control systems that are L^p well-posed are studied. Several interesting examples (such as the heat equation with Dirichlet control and Neumann observation) are however not L^p well-posed. To cover such examples in an abstract framework, I introduced distributional control systems. The frequency domain theory appears in (Opmeer 2005) and the time domain theory in (Opmeer 2006).

Control of flexible systems