7 papers
A Globally Asymptotically Stable Planar Homogeneous Polynomial Vector Field With No Polynomial Lyapunov Function
Jun Liu, Maxwell Fitzsimmons
We disprove the conjecture that every globally asymptotically stable homogeneous polynomial vector field admits a homogeneous polynomial Lyapunov function. The counterexample is a…
A Converse Control Lyapunov Theorem for Joint Safety and Stability
Thanin Quartz, Maxwell Fitzsimmons, Jun Liu
We show that the existence of a strictly compatible pair of control Lyapunov and control barrier functions is equivalent to the existence of a single smooth Lyapunov function that…
Rigorous Error Certification for Neural PDE Solvers: From Empirical Residuals to Solution Guarantees
Amartya Mukherjee, Maxwell Fitzsimmons, David C. Del Rey Fernández +1
Uncertainty quantification for partial differential equations is traditionally grounded in discretization theory, where solution error is controlled via mesh/grid refinement. Physi…
Safe and Robust Domains of Attraction for Discrete-Time Systems: A Set-Based Characterization and Certifiable Neural Network Estimation
Mohamed Serry, Maxwell Fitzsimmons, Jun Liu
Analyzing nonlinear systems with attracting robust invariant sets (RISs) requires estimating their domains of attraction (DOAs). Despite extensive research, accurately characterizi…
Computing Control Lyapunov-Barrier Functions: Softmax Relaxation and Smooth Patching with Formal Guarantees
Jun Liu, Maxwell Fitzsimmons
We present a computational framework for synthesizing a single smooth Lyapunov function that certifies both asymptotic stability and safety. We show that the existence of a strictl…
Symbolic Reduction for Formal Synthesis of Global Lyapunov Functions
Jun Liu, Maxwell Fitzsimmons
We investigate the formal synthesis of global polynomial Lyapunov functions for polynomial vector fields. We establish that a sign-definite polynomial must satisfy specific algebra…