A numerical method to solve matrix-valued differential inequalities with applications in dynamical systems

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A contraction metric is a tool in dynamical systems to show that the distance of adjacent trajectories decreases and thus to determine the long-term behaviour of solutions. A contraction metric is a matrix-valued function, and the contraction condition is a linear matrix-valued differential inequality. The project seeks to develop a numerical method to discretise such problems using meshfree collocation and then solve the resulting problem using semidefinite optimisation methods. It is expected that the solutions of the discretised problems will converge to the solution of the original problem. This general numerical method will then be applied to problems in dynamical systems such as the computation of a contraction metric. A main area of applications will be stochastic differential equations, where we will explore the use and computation of contraction metrics.The project has aspects of numerical analysis, functional analysis, stochastics, coding and applications to dynamical systems: we will prove convergence of solutions, write a code to solve differential inequalities, and apply it to the computation of contraction metric in dynamical systems, in particular for stochastic differential equations.The student will be co-supervised by Peter Giesl and Nicos Georgiou, forming a new, strong link between the Analysis and PDEs and the Probability and Statistics group in the Department of Mathematics at Sussex. Research visits of the PhD student to Holger Wendland (Bayreuth, Germany) and Sigurdur Hafstein (Reykjavik, Iceland) will complement the training at Sussex.