A Game-Theoretic Characterization of Feedback Capability for Fully Coupled Vector-Valued Nonparametric Systems

Abstract

We study feedback stabilization for the discrete-time system $x_{t+1}=f(x_t)+u_t+w_{t+1}$ in ${\mathbb{R}}^d$ with unknown $f$ and arbitrary bounded disturbances. For scalar plants, the sharp feedback capability threshold under generalized Lipschitz uncertainty is $3/2+\sqrt{2}$. We treat fully coupled vector-valued systems, where scalar order and interval recursion are unavailable and coupling precludes a coordinatewise reduction. We introduce a response-history escape game in which the adversary seeks a finite envelope and an unbounded state radius. Borel determinacy ensures that exactly one player has a winning strategy at each slope. We prove that the same player wins from every finite response history, and slope monotonicity gives an independently defined game value ${\Gamma }_d$. We prove that ${\Gamma }_d$ is finite and is the strict feedback capability threshold for the plant problem. If $L<{\Gamma }_d$, one causal feedback law stabilizes every plant in the uncertainty class against every bounded disturbance sequence. If $L>{\Gamma }_d$, for every causal feedback law there exist a plant in the same class and a bounded disturbance sequence such that the closed-loop state sequence is unbounded. The proof uses one controller for all subcritical slopes and a realization in a Hilbert space based on the Kirszbraun--Valentine extension theorem. An explicit nearest-neighbor law gives a lower bound above one in every finite dimension, including ${\Gamma }_2\ge 2/\sqrt{3}$. Dimension monotonicity gives ${\Gamma }_d\le {\Gamma }_1$, and comparison with the scalar theory yields ${\Gamma }_1=3/2+\sqrt{2}$.

22 pages