Quantitative local L₂-gain and Reachability analysis for nonlinear systems

This paper develops theoretical and numerical tools for quantitative local analysis of nonlinear systems. Specifically, sufficient conditions are provided for bounds on the reachable set and L₂ gain of the nonlinear system subject to norm-bounded disturbance inputs. The main theoretical results are extensions of classical dissipation inequalities but enforced only on local regions of the state and input space. Computational algorithms are derived from these local results by restricting to polynomial systems, using convex relaxations, for example the S-procedure, and applying sum-of-squares optimizations. Several pedagogical and realistic examples are provided to illustrate the proposed approach.