Optimizing Lyapunov Certificates via Stability-Preserving Quadratization for Polynomial Systems

Abstract

Region-of-attraction certificates for polynomial dynamical systems become computationally expensive as dimension and degree increase. We formulate optimal dissipative quadratization (ODQ), which jointly designs stabilizer gains and representation gauges for a fixed monomial lift and Lyapunov weight. For each gain, an exact semidefinite program minimizes a spectral-norm bound over the gauge, while residual-aware bounds account for floating-point Lyapunov residuals in the resulting certificate. Under the stated assumptions, accumulation points of the idealized outer search are box-Clarke stationary; global optimality of the gain search is not claimed. Experiments across heterogeneous polynomial systems demonstrate improvements over fixed-gain lifted baselines, while comparisons with direct sum-of-squares methods remain mixed.

Yubo Cai 蔡宇博

PhD student at MIT CCSE, CEE, and LIDS

Ph.D. student at MIT, jointly affiliated with CCSE, CEE, and LIDS. Research in optimization, system design, and control.