control system designers

Abstract: Sampling-based reachability estimates reachable sets from finitely many simulated trajectory endpoints. Existing guarantees often control missed probability mass or approximation error relative to the convex hull of the reachable set, leaving the geometric error relative to the reachable set incompletely characterized. We study this error as a problem of support estimation under flow distortion. Our assumptions combine a uniform lower bound on the local volume density of the initial set, a lower bound on the sampling density, and a one-sided Lipschitz constant $μ$ for the vector field. These conditions admit initial sets with corners and dynamics that may expand or contract. In the fine-accuracy regime, $\tilde{\mathcal O}\bigl(2^ne^{nμT}(R/r)^n/(ρκ)\bigr)$ independent endpoint samples suffice for Hausdorff accuracy $r$ using the endpoint cloud. Here, $n$ is the state dimension, $T$ is the horizon, $R$ is the volume-equivalent radius, and $ρ$ and $κ$ quantify sampling density and local volume density. For $n\ge2$ and sufficiently small $κ$, a minimax lower bound over all estimators based only on independent endpoints matches these dependencies up to a dimension-dependent factor $2\cdot4^n$ and logarithmic factors, within the stated parameter range and when the state domain accommodates the construction. The exponential burden in dimension, and in the horizon for expanding systems, is therefore intrinsic to this observation model. Planar experiments illustrate the predicted effects of corners and contraction; robot-arm experiments show how empirical accuracy deteriorates with dimension.

Abstract: Bisimulation metrics offer a principled notion of task-relevant state similarity in reinforcement learning, but existing constructions and methods are largely confined to discounted MDPs, where contraction arguments ensure well-defined fixed points and downstream guarantees. We develop a unified foundation for bisimulation metrics in undiscounted settings, that is total- and average-reward problems, by replacing the standard operator-centric view with a game-centric one: we interpret prior metric operators as Bellman-Shapley operators of zero-sum turn-based games and define metrics as the corresponding game value functions, yielding unique definitions even when $γ=1$. This game-theoretic foundation facilitates transfer of dynamic-programming theory and algorithms. We also use this game-centric perspective to characterize how metric finiteness depends on structural properties of the underlying MDP and motivate certain directed metrics that remain finite in weakly communicating MDPs. We further introduce an abstract framework for modifying the games' action spaces, allowing general transition coupling sets, that recovers many existing metrics and guides new constructions. Finally, we prove that these metrics bound total-reward and, when the optimal gain is state-independent, optimal relative value differences. We also provide error and suboptimality guarantees for aggregation-based planning in total- and average-reward settings.

Abstract: Bayesian optimization (BO) is typically used only for problems where the underlying function is deemed "expensive" (e.g. in time or money), with the speed of the BO algorithm being of little importance. In the real world, however, problems span a continuous range of values, going from cheap to expensive. In this paper, we present BO algorithms through the lens of a speed-performance Pareto frontier: we compare BO against faster algorithms like CMA-ES, and highlight how certain regions of the Pareto frontier are currently empty. As a solution, we propose a new set of fast BO methods that exploit the linear structure in existing BO algorithms, for both continuous and combinatorial inputs (from ~10 to 6000+ dimensions). Our methods achieve performance similar to state-of-the-art BO, while being more than $10^2\times$ to $10^3\times$ faster, opening the field of BO to an entirely new class of problems.

Abstract: Primal-dual methods for constrained Markov decision processes are analyzed under a two-timescale assumption, with non-stationarity entering through two channels. In the additive channel, where the constraint level oscillates, the dual oscillation saturates at a floor set by the moving optimum and is amplified linearly by the timescale ratio. In the parametric channel, where the constraint sensitivity oscillates, a coherent modulation carries no resonant forcing at any depth: the dual update's stiffness modulation cancels the primal update's damping modulation. Instability instead comes from a mismatch between the constraint estimate used by the primal update and the measurement used by the dual update, producing a parametric resonance at twice the pair's natural frequency. The tolerated mismatch grows with the square root of the forcing bandwidth and collapses with the square root of the dual-to-primal step-size ratio. These results hold on the tangent linear-quadratic reduction around an active saddle, for one constraint and for $K$ constraints under a common modulation and an isotropy condition. Driving both updates from the same estimate removes the resonant forcing but introduces a stationary offset equal to the estimator's bias. In a Lagrangian soft actor-critic agent, the multiplier locks onto the sub-harmonic at the timescale ratios its identified linear loop predicts and a coherent cost critic suppresses the lock-in, while a proportional (PID) term, predicted to leave the lock-in in place, suppresses most of it through the projection of the multiplier.

Abstract: In this short note, we prove that any closed interval in a well-ordered and discrete space is finite. The coincidence between finiteness and boundedness can then be established for transition systems where the state space is well-ordered and discrete. Finally, this result is also proven when the state space is a Cartesian product of $d$ discrete and well-ordered sets.

Abstract: The goal of the partial information decomposition (PID) is to quantify the redundant and synergistic information that multiple sources provide about a target. PID has many applications in machine learning, neuroscience, and other fields, but defining and computing it for high-dimensional continuous systems remains challenging. Here, we define a PID for multivariate Gaussian systems based on the Blackwell order, which formalizes when one channel is more informative than another. We prove that Gaussian channels are optimal for extracting both redundant and union information, yielding an intuitive geometric interpretation and an efficient numerical algorithm for the PID. Our union information and synergy coincide with the well-known BROJA measures, and we derive closed-form expressions for both in the case of two sources. We also argue that Blackwell redundancy (which differs from BROJA) is the only existing redundancy measure that satisfies a set of natural desiderata. We demonstrate the scalability of our method on systems with up to a thousand dimensions or sources. Our approach is illustrated on an optimal control problem, where it identifies redundant and synergistic interactions between sensor and memory.

Abstract: A multi-domain eXtended Physics-Informed Neural Network (XPINN) framework is developed for nonsmooth Delay Differential Equations (DDEs). This is the first implementation to demonstrate the efficacy of partitioning the temporal domain into subdomains of integer multiples of the characteristic time delay and progressively training the associated subnetworks while freezing previously learned parameters. The efficacy of the proposed framework is demonstrated using a machining dynamics model that incorporates both regenerative and nonsmooth frictional effects. Results demonstrate that the proposed multi-domain XPINN framework leads to better solution reconstruction in DDEs and improved parameter estimation compared to a generic PINN (SPINN) formulation. The proposed method works particularly well for extended temporal domains and non-constant history functions. The robustness of inverse XPINN (I-XPINN) is also assessed using reference data contaminated with Gaussian measurement noise. Results indicate that I-XPINN remains resilient to measurement noise and the physics-informed constraints guide the network toward accurately recovering the underlying dynamics. This demonstrates, for the first time, the potential of the proposed framework for reliable parameter identification in DDEs characterised by nonsmoothness and large time delays.

Abstract: We study nonconvex--strongly-convex bilevel optimization under a stochastic first-order oracle. We introduce MRT-FD, a single-loop first-order method that simultaneously tracks the upper-level variable, the lower-level solution, and the auxiliary response arising from implicit differentiation of the hyperobjective. MRT-FD performs one update of each variable per iteration and approximates the second-order derivative actions using order-$p$ finite differences. For any fixed finite smoothness order $p\ge1$ in the lower-level variable, MRT-FD finds an $\varepsilon$-stationary point using $\mathcal{O}(\varepsilon^{-4-2/p})$ stochastic gradient queries. We also prove a matching $Ω(\varepsilon^{-4-2/p})$ oracle lower bound. The lower-bound construction starts from a hard nonconvex minimization chain with a stronger stochastic oracle, and lifts it to a bilevel problem through a sinusoidal coupling with a scalar lower-level variable. Consequently, the dependence on $\varepsilon$ is optimal for every fixed finite $p$, closing the upper--lower complexity gap in this stochastic first-order oracle setting.

Abstract: Stochastic differential equations (SDEs) can be studied via Itô calculus and rough path theory. For stochastic optimal control, these two frameworks give distinct Pontryagin Maximum Principle (PMP) optimality conditions with forward-backward SDEs (FBSDEs) or rough differential equations. We show that the adjoint equations of the Itô and rough PMPs are connected via the conditional expectation $p_t^{\text{Itô}}=\mathbb{E}[p_t^{\text{rough}} \mid \mathcal{F}_t]$, where $\mathcal{F}_t$ represents information available at time $t$. First, we derive a rough stochastic PMP for problems with adapted controls that does not use FBSDEs. Its proof extends the rough stochastic PMP over deterministic controls by considering stochastic needle variations. Second, we derive a unified PMP connecting the Itô and rough PMPs, using Itô-Stratonovich conversion formulas and duality identities between the forward tangent and backward adjoint SDEs. As a first application, we rederive the adjoint matching method for fine-tuning generative models. As a second application, we propose an indirect shooting method for a class of feedback problems. Overall, these results give a new conditional bridge connecting two popular frameworks for stochastic optimal control.

Abstract: Classical stochastic approximation methods rely on estimators of the first moment (mean) of a random regression function. We study methods that employ estimators of both the first and the second moments, which include modern deep-learning optimizers such as Adam and Muon as special cases. We derive second-moment stochastic approximation methods through the lens of optimal preconditioning for solving matrix equations, and develop a two-stage framework for their convergence analysis. The first stage focuses on the analysis of conceptual (impractical) methods that rely on the exact first and second moments. In the second stage, we replace the exact moments with their respective estimators, and invoke Dvoretzky's theorem to show that the resulting practical methods converge almost surely to a neighborhood of the target solution. The size of the neighborhood depends on the biases and variances of the first- and second-moment estimators. We derive concrete bounds for Muon and a spectral variant of Adam that determine the radius of their neighborhood of convergence.

Abstract: Bayesian optimization (BO) is a widely used framework for optimizing expensive black-box objectives, but standard BO methods often use Gaussian process (GP) surrogates whose Gaussian assumption is sensitive to outliers and heavy-tailed noise. We introduce q-ED-BO, a robust BO method whose surrogate follows a univariate q-exponential (q-ED) distribution, preserving GP-BO's closed-form posterior mean and variance while a shape parameter q controls the tail behavior, recovering the GP at q = 2 and growing heavier-tailed with wider confidence bounds as q decreases. This tractability yields a closed-form q-upper confidence bound (q-UCB) with sublinear regret, and an exact closed-form q-expected improvement (q-EI) that generalizes EI to the heavy-tailed predictive, recovering classical EI at q = 2. Experiments on beamformer and adaptive filter tuning with impulsive outliers show that q-ED-BO matches or exceeds existing baselines on clean data, and under corruption, improves the strongest baseline by approximately 0.7 dB in output SINR and 1.1 to 1.2 dB in misalignment reduction.

Abstract: We show that unless P = NP, there cannot be a polynomial-time (or even pseudo-polynomial-time) algorithm for output feedback stabilization of a linear dynamical system with a linear controller. This settles one of the best-known open problems in control theory. The result holds in both continuous and discrete time. We also present a family of stabilizable linear dynamical systems for which no polynomial-time algorithm can write down a stabilizing controller in its standard representation.

Abstract: We propose a linearized Physics-Informed Neural Network (lPINN), a reduced-order neural basis method for forward and inverse differential equations. In an offline stage, lPINN learns operator-compatible continuous neural basis functions from an ensemble of numerical solutions. The basis functions are differentiable through automatic differentiation and are pretrained using solution data together with either derivative information or physics residuals. For each new problem instance, the basis functions are frozen and the solution is obtained by minimizing the governing-equation residual together with applicable initial, boundary, regularization, and observational terms. Unlike surrogate and operator-learning methods, the training data define the trial space offline, while the instance-specific solution is computed online by enforcing the governing physics. Relative to vanilla PINNs, lPINN pretrains the nonlinear hidden-layer representation offline and performs online inference only in the final linear layer. We evaluate lPINN on forward and inverse problems for the advection-diffusion equation, Burgers' equation, and the nonlinear pendulum equation. Compared with vanilla PINNs, lPINN achieves lower solution and parameter errors while reducing online inference times by approximately one to more than three orders of magnitude, with the largest gains generally observed for limited residual or measurement data. Cross-resolution experiments show that the learned continuous representation can be evaluated on finer meshes without retraining and with nearly unchanged accuracy.

Abstract: Assigning reduced coordinates to states near an attracting limit cycle requires the correct invariant-fibre geometry. The classical first-order phase-isostable chart obtained from adjoint Floquet modes projects along the strong-stable quotient fibre, whereas a metric-orthogonal complement of the retained slow bundle generally does not. We prove locally that a chart satisfying the linearised semiconjugacy relation leaves an O(delta^2) invariance residual, while projection along a non-invariant complement generically leaves an O(delta) term. For a nonlinear aeroelastic limit cycle, the metric-normal and strong-stable directions differ by 48.5 to 71.7 degrees, and metric-normal perturbations contain first-order retained phase and slow-amplitude components. Replacing the metric normal by the strong-stable fibre changes the measured residual scaling from delta^1.01 to delta^1.87 without fitted parameters. We then test learned higher-order corrections whose linearisation is pinned to the adjoint-Floquet chart, whose symmetry is exact, and whose reduced flow is fixed. Although they reduce the registered fixed-normalisation latent residual, post-hoc amplitude recalibration and adjoint-Floquet-targeted future consistency move or reverse the ranking. Because the learned maps already share the baseline's first-order gauge and the future target is supplied by the baseline chart, these diagnostics establish neither an independent positive nor negative higher-order result. Correct first-order Floquet geometry is therefore necessary in this benchmark, while the additional predictive value of the learned correction remains unidentified by the available representation-dependent diagnostics.