Numerical methods for differential Riccati equations

The aim of the proposed project is to analyze, develop and implement numerical methods for differential Riccati equations (DREs). These are matrix- or operator-valued evolution equations with quadratic nonlinearities. They are very important in many areas, in particular for optimal/robust control.

The solution to a DRE provides all optimal feedback laws for a linear quadratic regulator problem, and computing it is a critical step in the control of many industrial processes.There has been much interest in this area recently, with a focus on low-rank methods that can handle large-scale problems.

Many seemingly good methods have been proposed, but they all lack proper error analyses.

This critical gap means that it is difficult to tune them properly and to predict which problems they are most suited for.

It is also still not clear under which conditions matrix-valued DREs have solutions with low rank.A main goal of this project is therefore to perform rigorous error analyses of many methods by functional analytic arguments. We will focus on the operator-valued case and recover the matrix-valued case via spatial discretization.

Based on these insights, we will optimize existing methods, automate their usage for end-user engineers, and develop new advanced methods which combine their best features.

It will also lead to bounds on the rank of the exact solution.The research programme will be carried out over a 4-year period, by an associate senior lecturer assisted by a PhD student.