Analysis of systems described by partial differential equations using convex optimization

In this dissertation, computational methods based on convex optimization, for the analysis of systems described by partial differential equations (PDEs), are proposed.
Firstly, motivated by the integral inequalities encountered in the Lyapunov stability analysis of PDEs, a method based on sum-of-squares (SOS) programming is proposed to verify integral inequalities with polynomial integrands. This method parallels the schemes based on linear matrix inequalities (LMIs) for the analysis of linear systems and approaches based on SOS programming for the analysis of polynomial nonlinear systems.
Secondly, dissipation inequalities for input-state/output analysis of PDE systems are formulated. Similar to the case of systems described by ordinary differential equations (ODEs), it is demonstrated that the dissipation inequalities can be used to check inputstate/ output properties, such as passivity, reachability, induced norms, and input-to-state stability (ISS). Furthermore, it is shown that the proposed input-state/output analysis method based on dissipation inequalities allows one to infer properties of interconnected PDE-PDE or PDE-ODE systems. In this regard, interconnections at the boundaries and interconnections over the domain are considered. It is also shown that an appropriate choice of the storage functional structure casts the dissipation inequalities into integral inequalities, which can be checked via convex optimization.
Thirdly, a method is proposed for safety verification of PDE systems. That is, the problem of checking whether all the solutions of a PDE, starting from a given set of initial conditions, do not enter some undesired or unsafe set. The method hinges on an extension of barrier certificates to infinite-dimensional systems. To this end, a functional of the states of the PDE called the barrier functional is introduced. If this functional satisfies two inequalities along the solutions of the PDE, then the safety of the solutions is verified. If the barrier functional takes the form of an integral functional, the inequalities convert to integral inequalities that can be checked using convex optimization in the case of polynomial data. Furthermore, the proposed safety verification method is used for bounding output functionals of PDEs.
Finally, the tools developed in this dissertation are applied to study the stability and input-output analysis problems of fluid flows. In particular, incompressible viscous flows with constant perturbations in one of the coordinates are studied. The stability and inputoutput analysis is based on Lyapunov and dissipativity theories, respectively, and subsumes exponential stability, energy amplification, worst case input amplification and ISS. To the author’s knowledge, this is the first time that ISS of flow models is being studied. It is shown that an appropriate choice of the Lyapunov/storage functional leads to integral inequalities with quadratic integrands. For polynomial base flows and polynomial data on flow geometry, the integral inequalities can be solved using convex optimization. This analysis includes both channel flows and pipe flows. For illustration, the proposed method is used for input-output analysis of several flows, including Taylor-Couette flow, plane Couette flow, plane Poiseuille flow and (pipe) Hagen-Poiseuille flow.
We conclude this dissertation with a summary and an account for future research directions.

Keywords: Fluid Mechanics
Control Theory
Input-Output Analysis
Stability Analysis
Hydrodynamic Stability
Partial Differential Equations
Advisor: Papachristodoulou, A