We solve exactly a fundamental problem of adaptive control against adversarial disturbances: regulate the scalar system $x_{t+1} = ax_t + u_t + w_t$, $x_0=0$, $\|w\|_\infty \le 1$, where the constant pole $a \in [-Δ, Δ]$ is unknown in sign and magnitude and $Δ$ is arbitrarily large. Elementary as the system looks, the least worst-case peak $\|x\|_\infty$ that a causal controller can guarantee against an adversarial pair $(a, w)$ (the value of this game) has, to our knowledge, never been determined for any adaptive control problem with parametric uncertainty of arbitrary size under this criterion; existing theory supplies stability certificates, gain bounds, and regret rates, not the value. That value is $γ^\star(Δ) = 1 + Δ$ for every $Δ>0$. The summand $1$ is the irreducible price of the disturbance, and $Δ$ the exact price of a single, unavoidable identification spike. The optimal policy is certainty-equivalent deadbeat control at the midpoint of the set-membership consistent interval, an instance of the robust oracle $\times$ consistent model chasing architecture. The architecture is forced, not merely sufficient: writing $θ_t := -u_t/x_t$ exhibits every causal controller as an oracle-selector composition, and optimality pins the selector to the midpoint at the critical histories. The standard tools, classical and modern, each fail quantifiably: probing is punished before it pays, commitment is fatal at sub-disturbance excitation once adaptation is necessary, optimism degenerates to tie-breaking or pays asymptotically at least twice the optimum, and regret certificates are blind to the worst-case peak in both directions. The optimal law contains no exploration mechanism, its learning purely passive. These results give the first exact optimality certificate for consistent model chasing as a design principle for adversarial adaptive control.
Distribution steering seeks feedback laws that drive the state law of a dynamical system between prescribed initial and terminal distributions. Optimal transport provides a natural geometric approach, but its implementation generally requires a transport map or coupling in the full state space. Sliced optimal transport avoids this full-dimensional construction through one-dimensional projections. Yet, the resulting projected maps specify only directional displacements and do not by themselves prescribe a realizable feedback law. To this end, we develop a finite-horizon control framework based on sliced optimal transport. At each sampling instant, a projected optimal transport map defines a directional terminal condition, whose minimum-energy realization yields a randomized single-direction controller. Averaging over projection directions gives a deterministic sliced feedback. For the single-integrator dynamics, the averaged feedback makes the sliced Wasserstein distance to the target non-increasing. For Gaussian endpoint laws, it is affine, preserves Gaussianity, and steers the mean and covariance to their prescribed terminal values. We further identify a law-dependent gain that yields linear decay of the sliced Wasserstein distance together with an explicit characterization of the control energy. We also prove that the randomized controller converges to the averaged sliced flow as the sampling period vanishes. Finally, we extend the construction to linear dynamical systems. Reachability-normalized coordinates allow instantaneous realization of the sliced velocity for uniformly fully actuated systems, while local controllability Gramians provide exact finite-step realization for general controllable systems. Numerical examples illustrate the resulting distributional flows.
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