Nonlinear PDEs

Partial Differential Equations (PDEs) describe the locally averaged mass interaction of large numbers of constituent elements be they molecules, cells, organisms, or devices. Controlling such processes allows us to create lift over the wing of an aircraft, synthesise new chemicals and materials, and regulate combustion in power plants. The design of controllers for such processes, however, is difficult due to nonlinear interactions and a spatially distributed state. Existing methods either rely on techniques, such as lumping, to approximate the problem as an ordinary differential equation (ODE) or ad-hoc analytical steps, which cannot be leveraged in a widely applicable software package. 

The goal of this project is to develop efficient and automatic convex algorithms for the analysis of nonlinear PDEs, which can be implemented by combining the SOS and PIE frameworks. Furthermore, the project aims to develop a computational toolbox to enable specialists and non-specialists to harness the algorithms for analysis and controller design.