Finite-Horizon, Energy-Optimal Trajectories in Unsteady Flows

The paper empirically documents four key phenomena for finite-horizon MPC in the unsteady double-gyre: (i) a clear Pareto tradeoff between accumulated state error and actuation energy as R/Q varies (with figures directly labeling E=∑(x−xg)^T(x−xg)Δt and U=∑u^T uΔt), and discusses how this tradeoff shifts with horizon and gyre frequency ; (ii) energy spikes and scheduling around repelling FTLE ridges when crossing LCS, with visual correlation in Fig. 4 and text describing pre- and post-crossing actuation changes ; (iii) the frequent formation of periodic orbits around the goal and their dependence on gyre frequency ; and (iv) nonmonotonic cost vs. R/Q with bifurcations in trajectory shape, and similar bifurcations with horizon changes, often triggered by crossing LCS . These are presented as robust observations, not formal theorems. The model solution provides a rigorous optimal-control framing for the same setting: it proves that weighted-sum scalarization with positive weights yields Pareto-efficient points (consistent with the paper’s observed fronts), derives Pontryagin costate formulas linking control magnitude to the flow map derivative/Cauchy–Green tensor (explaining energy spikes near repelling LCS), and offers a principled bifurcation criterion based on crossing-energy thresholds governed by normal repulsion. It additionally refutes a stronger, non-paper claim about uniform additive near-optimality of short horizons, and proposes correct average-cost/discounted variants. Net: the paper’s empirical claims are supported, and the model’s theory is sound and complementary.