PDE Workshop

Location

LR7, Department of Engineering Sciences, University of Oxford, Parks Road,
Oxford, OX1 3PJ

 

Attendance Registation

Please click this link to the survey to register your attendance.

 

Summary

In recent years, several research groups have begun to investigate optimization and
Semi-definite Programming (SDP) methods for the analysis, control, and estimation of
Partial Differential Equations (PDEs). The development of efficient numerical techniques
has important applications across a range of fields, including: renewable energy,
robotics with flexible structures, synthesising new chemicals and materials, and many
others. In contrast to earlier approaches based on “lumping” or discretisation, these
emerging research efforts emphasise optimisation-based frameworks that enable
analysis and controller design directly at the infinite-dimensional level. The resulting
methods can provide rigorous guarantees of stability, robustness, and optimality, whilst
still capturing the effects of boundary inputs and avoiding the loss of physical properties.
The aim of this workshop is to bring together representatives from these emerging
research groups to exchange results, identify common challenges, and outline a shared
vision for the future development of this rapidly evolving field.

 

Schedule

Time	Event / Speaker
09.00 - 09.10	Opening Remarks
09.15 - 09.45	Emilia Fridman
09.50 - 10.20	Anton Selivanov
10.25 - 10.55	Christophe Prieur
11.00 - 11.30	Coffee Break
11.35 - 12.05	Matthew Peet
12.10 - 12.40	Sergei Chernyshenko
12.45 - 14.10	Lunch + Control Group Seminar
14.15 - 14.45	Declan Jagt
14.50 - 15.20	Giovanni Fantuzzi
15.25 - 16.00	Coffee Break
16.00 - 17.00	Discussion Session

 

Link to Project

Performance analysis of nonlinear PDEs using polynomial optimisation

 

 

Session Details

 

Emilia Fridman - Delay-Robust Control of PDEs

Some PDEs may not be robust with respect to arbitrary small time-delays in the feedback. Robust finite-dimensional controller design for PDEs is a challenging problem. In this talk two constructive methods for delay-robust control will be briefly presented:

1) Spatial decomposition (or sampling in space) method, where the spatial domain is divided into N subdomains with N sensors and actuators located in each subdomain;

2) Modal decomposition method, where the controller is designed on the basis of a finite-dimensional system that captures the dominant dynamics of the infinite-dimensional one. Sufficient conditions ensuring the performance and domains of attraction of the closed-loop systems are established in terms of simple LMIs that are always feasible for appropriate choice of controller parameters.

 

Anton Selivanov - L2 Residue Separation: A Framework for Spillover Avoidance in PDE Control

This talk presents a refined framework for designing finite-dimensional controllers for semilinear PDEs. The approach stabilises the dominant modes while accounting for the highly damped residual dynamics through the L2 input-to-residue gain. This perspective yields both quantitative improvements in performance bounds and new structural insights compared to existing truncation-based designs. Extensions to a broader class of systems are also discussed, including sampled-data implementations analysed via LMIs. Finally, potential extensions to nonlinear PDEs using sum-of-squares techniques are outlined. 

 

Christophe Prieur - Nonlinear Control of PDEs: A Constructive Method

This talk will review some recent works on the design of nonlinear control for the stabilization of PDEs. The class of PDEs will be very large, but we will focus on constructive methods of stabilizing controllers, as those providing explicit feedback gains. The controllers are usually boundary output feedback laws that could be either static or dynamic, with possibily the presence of some heteregoenities in the loop, or nonlinearities in the infinite-dimensional models.

 

Sergei Chernyshenko - Progress on bounds for dissipation in flows of general geometry reinforces the need for methods applicable to stochastic PDE

Choosing which approach (and there are quite a few) of using LMI in a PDE problem is not trivial. We will argue that, at least as far as bounds for infinite-time averages in fluid flows are concerned, the LMI approach applicability to stochastic PDEs is important. An explicit example of inviscid zero flow rate flow in a pipe under the action of non-zero mean pressure gradient along the pipe will be presented. Brief outline of the proof of the bound for energy dissipation in a flow through a pipe of general shape, as described in recent paper (Meccanica, 2025) will be given, on the base of which implications of the inviscid flow example explained.

 

Giovanni Fantuzzi - Semidefinite and moment-sum-of-squares methods for variational problems and PDEs

The development of semidefinite programming (SDP) methods and moment-sum-of-squares hierarchies for infinite-dimensional variational problems and partial differential equations has attracted growing interest in recent years. This talk will review recent advances in this area, highlight connections between seemingly disparate lines of work, and outline outstanding challenges in theory and computation.