2006/07/06 by Matthew M. Peet, Peet, Matthew M.
Computer Science · Engineering · Mathematics · #Dynamical Systems (math.DS) #FOS: Mathematics #Matrix Theory and Algorithms #Numerical Analysis (math.NA) #Numerical methods for differential equations #Stability and Control of Uncertain Systems
paper · pdf · doi:10.48550/arxiv.math/0607144
openalex publication_date 2006/07/06 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/01
This thesis addresses the question of stability of systems defined by differential equations which contain nonlinearity and delay. In particular, we analyze the stability of a well-known delayed nonlinear implementation of a certain Internet congestion control protocol. We also describe a generalized methodology for proving stability of time-delay systems through the use of semidefinite programming. In Chapters 4 and 5, we consider an Internet congestion control protocol based on the decentralized gradient projection algorithm. For a certain class of utility function, this algorithm was shown to be globally convergent for some sufficiently small value of a gain parameter. Later work gave an explicit bound on this gain for a linearized version of the system. This thesis proves that this bound also implies stability of the original system. In Chapter 7, we describe a general methodology for proving stability of linear time-delay systems by computing solutions to an operator-theoretic version of the Lyapunov inequality via semidefinite programming. The result is stated in terms of a nested sequence of sufficient conditions which are of increasing accuracy. This approach is generalized to the case of parametric uncertainty by considering parameter-dependent Lyapunov functionals. Numerical examples are given to demonstrate convergence of the algorithm. In Chapter 8, this approach is generalized to nonlinear time-delay systems through the use of non-quadratic Lyapunov functionals.