CDS THESIS SEMINAR: Scalable Analysis of Nonlinear Systems Using Convex Optimization

Antonis Papachristodoulou, Caltech, Control and Dynamical Systems

There is a noticeable asymmetry between the richness of available frameworks for modelling physical systems and algorithmic analysis methodologies for such models at different scales. Moreover, technological advances and the integration of computer, communication and control pose new challenges on guaranteeing scalable functionality of large-scale system interconnections of subsystems described by ordinary or even delay differential equations.

In this work, we investigate how convex optimization can be used to analyze systems described by different classes of nonlinear differential equations in an algorithmic fashion. The methodology is based on the construction of appropriate Lyapunov-type certificates using sum of squares techniques. Motivated by a series of examples from engineering and biology we develop theory to answer robust stability and performance analysis questions algorithmically for systems described both by finite (ordinary) and infinite-dimensional (delay) differential equations.

We then concentrate on scalable stability analysis of congestion control algorithms for the Internet. The modelling framework we use results in an arbitrary interconnection of subsystems described by delay differential equations. We concentrate on FAST, a recently developed network congestion control scheme, for which we construct a Lyapunov certificate so that the Lyapunov properties scale with the system size, thus ensuring that stability holds for arbitrary delays, network topologies and link capacities. The structure of the system plays an important role in the Lyapunov construction. The same technique can be used to analyze other network congestion control schemes.