Generalizable Optimal Control with Transformers: Closed-Loop Certification and Near-Optimality Guarantees

This letter develops closed-loop performance certificates for a transformer-based feedback policy. The policy is trained to imitate optimal Linear Quadratic Regulator (LQR) control across a family of heterogeneous Multiple-Input, Multiple-Output (MIMO) Linear Time-Invariant (LTI) systems. First, we establish a finite-sample excess-risk bound for the imitation loss minimized during training. Second, for each fixed problem instance, we derive regional closed-loop guarantees consisting of a forward-invariant operating region and a worst-case bound on deviation from the optimal rollout. Our main result is a probabilistic certificate for finite-horizon closed-loop near-optimality. Using an exact LQR cost identity, we express excess cost as a measurable per-rollout statistic and use independent calibration and validation rollouts to obtain a high-confidence bound on its violation probability. We evaluate the certificate on $28$ benchmark systems. This uses the base policy on seen systems and system-specific fine-tuned copies on unseen systems, with each rollout drawing the plant, cost, and initial condition from the corresponding certification distribution. All per-system certificates have violation probabilities below $3.1\%$, each at $95\%$ confidence; twenty systems certify suboptimality below $10\%$, with the tightest threshold equal to $4.8\times10^{-6}$.