CONTROL ANALYSIS AND SYNTHESIS FOR GENERAL CONTROL-AFFINE SYSTEMS

The paper proves global exact controllability for ẋ = A(t,x)N_t(x) + B(t,x)u + f(t,x) by freezing the x-dependence along an arbitrary path z, defining a frozen flow Φ^z and Gramian-like matrices W1^z, and then synthesizing an implicit control u_z via N1^z(u_z) built along the controlled trajectory x_{u_z}^z(·). It then constructs a compact, continuous self-map F(z) = x_{u_z}^z and applies Schaefer’s theorem to obtain a fixed point and hence x(T)=x1 (see eqs. (3.10)–(3.15) for N^z, W^z, the control formula (3.20), and the fixed-point construction around (3.19)–(3.23), culminating in Theorem 3.6) . The core identity comes from the solution representation of the frozen system, in which the pullback integral depends on DΦ^z evaluated along the controlled state x(·), not along the drift-only path (Theorem 3.2) . By contrast, the model’s solution incorrectly asserts that the endpoint map is affine in u with kernel M_z(t)=DΦ^z_{t,t0}(Φ^z_{t0,t}(x0))B^z(t) and thus constructs an explicit right-inverse using W1^z(T); this replaces DΦ^z(·) evaluated at x(t) by its value along the drift path, which is not valid for the nonlinear system. The paper addresses precisely this nonlinearity by using the u-dependent matrices N1^z(u) and a fixed-point argument; the model’s linearization step and the claim of an affine endpoint map are therefore wrong (cf. (3.10)–(3.15) vs. the model’s (∗) and ensuing construction) . Additionally, the model applies Schauder to a map T defined via the original dynamics without ensuring uniqueness/continuity; the paper avoids this pitfall by freezing A,B,f to guarantee global Lipschitzness and a single-valued map F (Lemma 3.1 and the compactness/continuity estimates (3.21)–(3.23)) .