Learning Koopman Representations for Hybrid Systems

The candidate reproduces the paper’s Theorem 2.1 construction almost verbatim: introduce v_k := x_{k+1} − x_k, define control/discrete increments φ_k and ω_k, assume the single-step identifiability condition so that G(·,x,u): V(x,u) → S is a bijection, and evolve a continuous-state, time-varying system with state s_k = (x_k,u_k,v_k) ∈ R^{2n+m} that is in bijection with (x_k,y_k) along the fixed {u_k,z_k} trajectory. This is precisely the paper’s setup and proof (Eqs. (16)–(32)), including the definition of V(x,u), the decoder via G yielding y_k and z_k, and the update for v_{k+1} (Eq. (29)); the paper explicitly concludes the bijection “Given u_k and z_k” between (x_k,y_k) and s_k (x,u,v) . The candidate’s only addition is to make the encoder/decoder (E_k,H) explicit and to verify invertibility by induction, which is consistent with and not stronger than the paper’s argument.