- Functional analysis
- Control theory
- Partial differential equations
- Numerical analysis
- Complex analysis

## Research themes

My current research can be roughly grouped into a number of themes:- Infinite-dimensional Lur'e systems
- Model reduction
- Linear Quadratic optimal control
- Numerical methods for Lyapunov, Riccati and Lur'e equations

- State space methods in the factorization approach to controller design
- Distributional control systems
- Control of flexible structures

### 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.

- My earliest work in this area (Opmeer and Curtain 2004, Opmeer 2007) generalized LQG-balancing (including error-bounds in the gap metric) to infinite-dimensional systems.

- Later work considered Lyapunov-balanced model reduction. In (Guiver and Opmeer 2014) the well-known error bound for Lyapunov-balanced model reduction was generalized to general infinite-dimensional systems. A lower-bound for the Lyapunov-balanced error was given in (Opmeer and Reis 2015). A detailed analysis of the error (meaning the decay rate of the Hankel singular values) was carried out in (Opmeer 2010) and (Opmeer 2015).

- Three articles written jointly with Chris Guiver consider dissipativity preserving balanced model reduction: A gap metric error bound is proven for the finite-dimensional case in (Guiver and Opmeer 2013, Linear Algebra and its Applications) and for the infinite-dimensional case in (Guiver and Opmeer 2013, Mathematical Control and Related Fields). The article (Guiver and Opmeer 2011) contains a counter-example to an error bound for dissipativity preserving balanced model reduction proposed by others in 1995.

- In the article (Opmeer 2012) a connection is made between model reduction by rational interpolation and model reduction by balanced proper orthogonal decomposition.

### 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.

- In (Curtain and Opmeer 2005) state space formulas for a solution of the suboptimal Nehari problem were obtained under very moderate stability assumptions. These were used in (Curtain and Opmeer 2006) to obtain state space formulas for Bezout factors under very moderate stabilizability assumptions. State space formulas for the resulting dynamic robustly stabilizing controller were subsequently obtained in articles by my co-author Ruth Curtain. This was all for the continuous-time case. The corresponding discrete-time results are in (Curtain and Opmeer 2009) for the Nehari problem and in (Curtain and Opmeer 2011) for the Bezout factors and dynamic robustly stabilizing controllers.

### 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

- In (Guiver and Opmeer 2010) we show that a non-dissipative boundary feedback that was known to stabilize an Euler-Bernoulli beam model is actually destabilizing for Rayleigh and Timoshenko beam models.
- In (Opmeer 2011) I considered infinite-dimensional systems with force control and position measurement. The finite-dimensional was earlier studied by others in the context of negative imaginary systems and the mentioned article is a first step to connect this theory to the theory of infinite-dimensional systems.