Multi-Objective LQR with Linear Scalarization

We study finite approximation of the Pareto front for the multi-objective linear quadratic regulator (LQR). Classical results characterize Pareto-optimal controllers through weighted sums of the objectives, but do not quantify the effect of discretizing the scalarization weights. We prove the needed sensitivity property, showing that an $\epsilon$ perturbation of the weights changes the full objective vector by $O(\epsilon)$. Consequently, computing optimal controls over a finite grid of the weight simplex uniformly approximates the Pareto front. We also extend the guarantee to unknown dynamics via certainty equivalence, showing that if the model estimates are accurate to the target accuracy, the same procedure stabilizes the true system and attains the same Pareto-front guarantee.