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 $\Gamma_1=3/2+\sqrt{2}$.