Fundamental Limits for Sensor-Based Control via the Gibbs Variational Principle

PAC-Bayesian bounds provide finite-sample guarantees for data-dependent randomized predictors, but applying them to learning-based control is difficult because the natural objective is a quadratic trajectory cost. Such losses are unbounded, non-Lipschitz , and lead to response-dependent Chernoff terms. We employ System Level Synthesis parameterization, which exposes the closed-loop trajectory map of a linear system directly and makes the quadratic control loss amenable to explicit certification. Moreover, we provide a set of PAC-Bayes-Chernoff certificates for posterior distributions over feasible closed-loop responses. For Gaussian disturbance trajectories with arbitrary covariance, we derive an exact one-sided Gaussian transform and a tractable quadratic upper bound expressed through closed-loop sensitivity quantities. We also derive a posterior-localized surrogate for settings where pointwise closed-loop response certificates are unavailable or have support related admissibility issues. Although PAC-Bayes certifies a non-degenerate posterior, the convex quadratic form of the SLS loss transfers the certificate to the posterior mean response. We present a deterministic mean response deployment result that is particularly suitable for control while retaining the stochastic posterior in the bound. Additionally, we provide a data-driven bound for this deployment, transitioning away from an oracle bound. Minimizing this bound naturally results in a learning algorithm for control selection from data. Numerical experiments on a double integrator show that the algorithm acts as a sensitivity-aware finite-sample regularizer, improving held-out cost and reducing closed-loop sensitivity in the low-data regime

This paper investigates the control of discrete-time linear time-invariant (LTI) systems subject to incomplete and corrupted measurements. Specifically, we focus on designing a Linear Quadratic Gaussian (LQG) controller without relying on explicit state estimation. By leveraging minimum variance duality, our approach allows the current control input to be represented as a linear function of available measurements and previously applied inputs, successfully reducing the task to a tractable deterministic optimization problem. We provide theoretical justification for this framework and demonstrate its practical effectiveness through numerical experiments.

Finite-horizon optimal-control computations repeatedly solve two-point Pontryagin boundary value problems whose conditioning can deteriorate as the horizon grows. We give a verifiable data-level certificate under which it does not. Hyperbolicity of the reduced state--costate transition matrix, together with scaled stable--unstable boundary transversality, yields an endpoint-corrected Green inverse with horizon-independent constants and weighted contractions transfer this inverse to the nonlinear problem, so the original Pontryagin endpoint rows $x_0=x_{\rm in}$ and $p_T=r_x(x_T,y)$ carry a unique local stationary branch whose first-order expansion and Lipschitz constants are uniform in the horizon. Consequently the finite-horizon feedback map is horizon-uniformly Lipschitz, first-order expandable, and satisfies an exact shrinking-horizon consistency identity. Symplectic and Riccati criteria certify the hypotheses from matrix data: every stabilizable definite linear-quadratic system with invertible dynamics and a locally concave terminal Hessian at the reference qualifies. Reproducible computations illustrate both certificates.