Regret-optimal control in dynamic environments

The paper derives the regret-optimal controller for finite-horizon LTV LQR by reducing the regret-suboptimal inequality to a suboptimal H∞ problem via a causal, invertible whitening of the disturbance, z = Lw, where L is constructed so that γ^2 I + G^T (I+FF^T)^{-1} G = L^T L; the resulting extended plant, Riccati recursions, and strict bounded-real conditions Δ̂_t ≺ 0 yield the controller in Theorem 4 and the bound Regret ≤ γ_opt^2 ∑ ||w_t||^2 (see the reduction and state-space model for z and the extended plant, and Theorem 4’s formulas for Â_t, B̂_{u,t}, B̂_{w,t}, Q̂_t, P̂_t, P^b_t, and Δ̂_t) . The candidate solution reproduces this structure step-by-step: forward completion-of-squares to define Ã_t and the disturbance-shaping state δ via a forward Riccati, a backward completion-of-squares and change of variables w ↦ z via a backward Riccati (R^b_{e,t} and K^b_{l,t}), construction of the same extended plant, and enforcement of the finite-horizon bounded-real inequalities Δ̂_t ≺ 0 with synthesis via dynamic programming. The H∞ subproblem and bisection over γ^2 match the paper’s Theorem 2-based approach (suboptimal H∞ feasibility via Δ_t ≺ 0) and Theorem 4 (smallest γ^2 with Δ̂_t ≺ 0), respectively . Aside from stylistic differences (operator/Kalman-factorization viewpoint in the paper vs. completion-of-squares/dissipation in the candidate), the definitions and resulting controller are the same.