Reachability of Nonlinear Systems with Unknown Dynamics

The paper’s Theorem 4 proves that the proxy control system ẋ = a + UΛ(‖x‖)u with λ_i(s) = max{ g(s)/(α(s)‖G(0)†η_i‖ + β(s)), g(s), 0 } has reachable set R̂(T,x0) contained in the guaranteed reachable set RG(T,x0), by embedding a polygon P(S(x)) inside the guaranteed velocity set VGx and invoking Proposition 1. The candidate solution establishes the same inclusion by showing the pointwise velocity-set inclusion a + UΛ(‖x‖)u ∈ f̂(x) + Ĝ(x)B_m(0;1) for every consistent (f̂,Ĝ), using Weyl-type singular-value bounds, pseudoinverse perturbation bounds with a shape constant μ, and a minimal-norm control construction; this matches the paper’s construction and constants and reaches the same conclusion. Apart from minor clarity issues (notation for the input subspace and a typographical “‖G(0)†‖ vs ‖G(0)†‖−1” in the domain bound), the arguments are consistent.