The problem of optimal sizing and power scheduling in microgrids subject to uncertainties is well known to the control community. Commonly, the optimal control problem is cast as a mixed-integer program to model the logical constraints arising in energy storage systems, and is then solved approximately using numerical methods such as the scenario approach. In this paper, we propose and compare two formulations of a robust microgrid sizing and power scheduling optimal control problem with logical constraints and uncertainties in the user's power demand, solar power generation, grid electricity prices and battery efficiencies. The first formulation uses binary variables and big-M constraints, leading to a mixed-integer linear program. The second formulation casts the problem as a continuous nonlinear program through an exact smooth reformulation of the logical constraints, consisting of additional modelling variables and non-convex constraints. We then propose a novel local reduction algorithm, extending an existing method, to solve both problems. The two formulations are compared by evaluating the solutions returned by local reduction using 100,000-sample Monte Carlo simulations and achieve promising results, with both averaging feasibility rates above 90%.

We develop a large-signal stability analysis for a sampled-data, optimization-based secondary controller for inverter-interfaced distributed energy resources in virtual power plants.

This note is a tutorial on the deterministic version of the Kalman filter (state estimator), which is formulated as finding the state trajectory consistent with the system's equations with the minimal amount of $L^2$ process and measurement uncertainty. As stated, this is an input signal design problem with linear dynamics and an objective that is affine-quadratic in the state and inputs. The first step is to convert this problem to one with a purely quadratic objective by embedding in a larger system using ``homogeneous coordinates''. This converts the problem to a purely quadratic (i.e. an LQR) problem, but with non-standard initial or final state constraints. This latter problem can then be solved using a version of the matrix Differential Riccati Equation (DRE) for the larger LQR problem. The second step is a partitioning of this larger problem, which then yields the optimal dynamic observer and the DRE of the traditional Kalman filter. For comparison, the solution of the traditional LQ-tracking (Servomechanism) problem is also treated using a similar construction.

In this paper, we investigate a stochastic averaging principle for a large class of mechanical systems in the presence of random vibrations. We show that a known deterministic averaging principle for affine connection control systems with large-amplitude high-frequency inputs also holds for non-periodic stochastic inputs. The randomly vibrating mechanical system is described by a stochastic differential equation whose solutions do not depend on its interpretation in the sense of Ito, Stratonovich, or Sussmann. We also show that solutions of this stochastic differential equation can be directly computed from a single ordinary differential equation. We illustrate our theoretical results by the example of stochastic source seeking with a nonholonomic vehicle.

Identifying the governing equations of complex dynamical systems remains a fundamental challenge across science and engineering. While early approaches relied on empirical data and heuristics, modern data-driven methods offer greater flexibility and fewer assumptions. However, data acquisition in real-world settings is often expensive. This work addresses this challenge by introducing an active learning strategy for dynamics discovery in the ultra-low data limit. Rather than sampling randomly, our method iteratively prioritizes regions that are most informative for model identification. This approach builds on Sparse Identification of Nonlinear Dynamics (SINDy), and utilizes an ensemble extension, E-SINDy, to estimate epistemic uncertainty and guide the sampling for both ordinary and partial differential equations (ODEs/PDEs). For ODEs, an exhaustive analysis is conducted on the Lorenz system across varying data budgets and noise levels. For PDEs, two systems with contrasting dynamical characteristics are examined: the Burgers' equation, where a sharp shock front creates a distinction between informative and uninformative regions, and the Kuramoto-Sivashinsky equation, which presents a more spatially complex sampling landscape. Across all scenarios, the proposed method accurately identifies the governing dynamics with significantly fewer data samples than random sampling.

Moving horizon estimation (MHE) provides high precision state estimation for nonlinear systems, but it is often limited by the substantial computational demands of solving a nonlinear optimization problem at every sampling step. To address this issue, we develop an efficient MHE scheme based on linear parameter-varying (LPV) formulation, where the scheduling parameters are given by the estimated states of the system and used to construct inexact Jacobians. Due to the LPV representation, the Jacobian can be pre-specified offline in a structured form and then updated in the quadratic programming (QP) subproblem, which reduces computational cost commonly used in standard nonlinear programming (NLP) systems. We illustrate the performance by numerical simulations.

The problem of pricing mobility services has attracted significant attention. In most studies, uncertain demand is modeled as an exogenous random variable with known distribution. This assumption overlooks the likely effect of prices on user adoption decisions. To address this dependency, we formulate the pricing problem as a stochastic program with decision-dependent demand uncertainty. Specifically, we make the non-standard assumption that the probability distribution of demand depends on pricing decisions. We show that the problem can be written as a mixed-integer linear program whose size is exponential in the input parameters. To find exact numerical solutions we specialize the L-shaped method for stochastic programs with decision-dependent uncertainty. In particular, we devise efficient separation routines by proving closed-form primal and dual solutions to the involved subproblems. In addition, we develop problem-specific valid inequalities and cut-sharing mechanisms which significantly improve convergence. We show that the method outperforms by far a commercial solver used to solve the monolithic formulation. Furthermore, in a case study based on a real-world carsharing system, we show that incorporating decision-dependent uncertainty improves expected profits by 8.39% compared to a benchmark that considers deterministic price-elastic demand, and by 8.53% compared to a benchmark that considers exogenous random demand, on average. In addition, we evaluate the performance of preventive pricing and relocation decisions under two vehicle allocation policies. The results suggest that a controlled allocation of vehicles to customers can improve service rates while only marginally affecting profits.

Port-Hamiltonian systems provide a structured framework for modeling physical systems by explicitly capturing their energy storage, dissipation, and exchange. However, deriving such models often requires detailed physical insight and precise knowledge of system parameters, which may not be available in practice. In this paper, we propose a kernel-based framework for the identification of port-Hamiltonian systems from input-state-output data. In contrast to conventional parametric approaches, the maps defining the port-Hamiltonian system are represented in suitably chosen reproducing kernel Hilbert spaces. This leads to an infinite-dimensional optimization problem over the corresponding function spaces. Our main result establishes a representer theorem that reduces this problem to a tractable finite-dimensional one. Since the reduced problem is non-convex, we further provide an algorithm for its solution and prove its convergence.

This paper develops a fully discrete cumulative distribution transform (CDT) for atomic probability measures on the real line. The transform is defined through monotone quantile maps and admits explicit linear-time algorithms for both forward transformation and inverse reconstruction based solely on cumulative mass matching. Unlike the classical continuous setting, deterministic transport between atomic measures cannot generally split masses, so exact reconstruction may fail at finite resolution. We establish a precise cumulative-mass compatibility criterion for exact finite-resolution recovery and prove weak convergence of reconstructed measures under reference refinement. Several structural properties of the discrete CDT are derived, including translation, composition, and scaling laws, and the framework is extended to a discrete signed cumulative distribution transform with thresholded stabilization near zero crossings. By avoiding continuous interpolation, the proposed framework provides a simple fixed-reference transport representation for discrete data. Numerical examples illustrate translation linearization, compatibility-controlled reconstruction, refinement consistency, and stabilization of the signed transform.

We propose ZOMA, a unified Zeroth-Order decentralized accelerated MinimAx framework for multi-agent nonconvex Polyak--Łojasiewicz minimax optimization. The proposed framework only requires evaluating the function value and, as such, is tailored to gradient-free environments, where exact gradient information is either unavailable or computationally prohibitive to obtain. A central contribution of our \textbf{ZOMA} framework is a multi-level unification, along the following directions: (i) \emph{estimator} - our framework adopts a hybrid zeroth-order estimator, which accommodates, among others, both coordinate-wise and randomized uniform smoothing estimators; (ii) \emph{bias correction} - our framework subsumes a wide range of bias-correction strategies, including gradient tracking (GT), exact diffusion (ED), and EXTRA and (iii) \emph{acceleration} - our framework facilitates a broad class of acceleration techniques, including zeroth-order versions of STORM, PAGE, and L2S. The general nature of \textbf{ZOMA} leads to many novel decentralized zeroth-order minimax methods and allows us to establish unified convergence guarantees, matching the performance of state-of-the-art centralized zeroth-order minimax methods, while providing benefits, such as linear speed-up in the number of users. The unified framework also provides a systematic way to assess algorithmic suitability by specializing the convergence rates to specific problem structures and method designs. We validate the performance of the proposed algorithms via numerical simulations.

Low-rank optimal transport (OT) mitigates the quadratic scaling of classical solvers, yet existing approaches rely heavily on first-order mirror-descent updates that require careful hyperparameter tuning and ignore the optimization landscape's curvature. To address these limitations, we propose a unified Riemannian geometric framework for low-rank OT, modeling balanced and unbalanced rank-$r$ positive factored couplings as novel smooth embedded submanifolds of the positive orthant. By equipping these manifolds with the Fisher-Rao product metric, we derive tractable formulations for Riemannian projectors, retractions, and Hessian-vector products. Our cost-agnostic framework seamlessly extends to linear OT, Gromov-Wasserstein (GW), fused GW, and their unbalanced counterparts. For balanced OT, our geometric ingredients are computed via efficient conjugate-gradient and iterative Bregman updates. For the unbalanced OT, our operations elegantly reduce to closed-form scalings, completely eliminating inner iterative loops. In both regimes, per-iteration complexity scales linearly with dataset size, and we provide a rank-sufficiency certificate for global optimality verification. Extensive experiments across a range of problem sizes demonstrate that our regularization-free first- and second-order solvers achieve faster convergence and superior performance over existing state-of-the-art low-rank OT solvers.

Lyapunov feedback laws can be constructed for performing orbital transfer maneuvers based on Gauss Variational Equations, or GVEs, while satisfying specified state and control constraints. These state and control constraints are enforced using barrier functions and saturation, respectively, while the reference governor, employed as a convergence governor, is used to avoid getting stuck at spurious equilibria that may be created by barrier functions. In this article, these Lyapunov feedback laws are extended to rendezvous maneuvers where not only five orbital elements are matched to prescribed target values, but also the true anomaly matches its time dependent target value. The modification involves altering the commanded semi major axis with an outer loop feedback law, also designed using Lyapunov techniques. We illustrate the resulting safe closed loop rendezvous maneuvers in simulations for conventional thrust based propulsion and Lorentz force-based propulsion. In the latter case, only the current through the tether is controlled, subject to current limits, to accomplish prescribed orbital transfer and rendezvous maneuvers.

Community detection in complex networks is frequently challenged by incomplete or noisy adjacency matrices. Traditional symmetric nonnegative matrix factorization methods typically rely on zero-imputation for unobserved entries, which compromises clustering reliability. This paper proposes a Masked Symmetric Nonnegative Matrix Factorization (Masked SymNMF) framework designed to factorize partially observed networks directly. By defining a masking operator over the observed entries, the proposed model restricts the objective evaluation exclusively to valid data. To overcome the severe non-convexity inherent in the symmetric factorization, we formulate an asymmetric relaxation penalized by a regularization term. We prove the exact penalty property of this reformulated model, establishing its theoretical equivalence to the original symmetric problem under sufficient regularization. Furthermore, an alternating nonnegative least squares framework is developed, yielding tailored update rules for Multiplicative Updates, Hierarchical Alternating Least Squares, and Projected Gradient Descent algorithms. Extensive numerical experiments on synthetic datasets and real-world networks demonstrate that the proposed Masked SymNMF outperforms baseline imputation methods across varying observation densities, providing a theoretically sound and practically efficient approach for community detection in incomplete networks.

This paper presents a collaborative adaptive formation control framework for autonomous vehicles (AVs), that explicitly handles system uncertainties, input saturation, and communication delays. To overcome the inherent physical torque limits of steering and braking actuators, an input saturation compensation mechanism is introduced to render nonlinearities tractable and improve control reliability. Additionally, a delay-compensating auxiliary system is designed to mitigate the effects of communication delays and reduce tracking errors. Our framework incorporates a dynamic-threshold event-triggered control (ETC) strategy to optimize resource usage. Additionally, uncertainty observers and symmetric barrier Lyapunov functions are developed to ensure robust and safe formation maneuvers. Finally, the effectiveness of the proposed approach is validated through numerical simulations of vehicle formations, complemented by a 3D visualization video demonstrating the dynamic fleet reconfiguration process.

In this paper, we introduce a class of preference robust risk measures-\emph{robust optimized certainty equivalents} (ROCE)-which encompasses several widely used measures, including Conditional Value-at-Risk and expectiles, as special cases. Motivated by recent developments in distributionally robust optimal reinsurance (DROR), we investigate DROR problems under the ROCE risk measure and consider two prominent uncertainty sets: the mean-variance uncertainty set and the Wasserstein uncertainty set. For the mean-variance uncertainty set, we reformulate the infinite-dimensional optimization problem into a finite-dimensional one by showing that it suffices to consider three-point distributions. This leads to a unified and explicit formulation for a broad class of ROCE risk measures and offers a simplified framework that also recovers earlier results for Conditional Value-at-Risk and expectiles. For the Wasserstein uncertainty set, we also derive a tractable finite-dimensional formulation. The resulting data-driven models enable efficient computation and facilitate a systematic comparison between moment-based and Wasserstein-based uncertainty sets in the optimal deductible design. Numerical experiments are exhibited to illustrate the performance of our reformulated programs.

For a set $X$ of integer points, the relaxation complexity $\operatorname{rc}(X)$ is the smallest number of facets of any polyhedron $P$ such that $P \cap \mathbb{Z}^d = X$. In this paper, we focus on the case where $X$ is the discrete standard simplex $\Delta_d = \{\mathbf{0}, \mathbf{e}_1, \dots, \mathbf{e}_d\}$. We show that $\operatorname{rc}(\Delta_d) = O(\log d)$ by an explicit, elementary construction. This improves upon the previously best-known upper bound $\operatorname{rc}(\Delta_d) = O(d / \sqrt{\log d})$ due to Aprile, Averkov, Di Summa, and Hojny (2024) and matches an asymptotic lower bound by Averkov and Schymura (2022).

We consider linear programming in the oracle model: $\max\{c^\top x \,:\, x\in P\}$, where the polyhedron $P=\{x\in\mathbb{R}^n\,:\, Ax\le b\}$ is given by a separation oracle. We present an algorithm that finds exact primal and dual solutions using $O(n^2\log(n/\delta))$ oracle calls and $O(n^4\log(n/\delta)+n^5\log\log(1/\delta))$ arithmetic operations, where $\delta$ is a geometric condition number associated with the system $(A,b)$. These bounds do not depend on the cost vector $c$ and do not require a priori knowledge of $\delta$. For rational data, $\log(1/\delta)$ is polynomially bounded in the encoding size of $(A,b)$, thus providing a polynomial-time algorithm. The algorithm works in a black box manner, requiring a subroutine for approximate primal and dual solutions; the above running times are achieved when using the cutting plane method of Jiang, Lee, Song, and Wong (STOC 2020) for this subroutine. Whereas approximate solvers may return primal solutions only, we develop a general framework for extracting dual certificates based on the work of Burrell and Todd (Math. Oper. Res. 1985). Our algorithm strengthens results by Grötschel, Lovász, and Schrijver (Prog. Comb. Opt. 1984), and by Frank and Tardos (Combinatorica 1987) that rely on bit-complexity arguments. Our algorithm avoids rounding-based arguments such as simultaneous Diophantine approximation and uses geometric arguments instead.

Convex optimization problems with linear equality constraints arise ubiquitously in scientific computing, machine learning, and control theory. Classical Krylov methods are effective but rely on problem-specific preconditioners and high memory. Conversely, gradient-based methods like the augmented Lagrangian method (ALM) avoid these issues yet suffer from slow outer iterations. Developing accelerated outer-iteration schemes, therefore, remains a critical research objective. In this study, we demonstrate that incorporating a proximal operation into the augmented Lagrangian framework yields the proximal ALM, where the outer iteration is equivalent to an implicit gradient descent-ascent (GDA) scheme. We further establish that this equivalence extends naturally to the setting of variable step sizes. Through Lyapunov analysis, we show that the underlying potential function must be shifted from the conventional objective gap to a variational inequality measure, signaling a shift in perspective from pure convex optimization to minimax optimization. Motivated by these observations, we first develop an implicit GDA scheme with variable step sizes based on a continuous-time ODE framework, which achieves an $o(1/k)$ last-iterate convergence rate for both the primal-dual objective gap and the gradient norm. Building upon a second-order ODE framework, we then propose a family of Nesterov-type implicit GDA schemes parameterized by $r \geq 0$, which achieves an $o(1/k^{r+1})$ last-iterate convergence rate for the primal-dual objective gap. Furthermore, specializing the second-order ODE formulation to the case $r=0$, we derive a corresponding explicit GDA scheme and prove an $o(1/k)$ last-iterate convergence rate for the primal-dual objective gap. Finally, we present several numerical experiments to validate these theoretical results and demonstrate the effectiveness of the proposed methods.

In this paper, we develop a continuous-time model-free reinforcement learning algorithm to learn deterministic equilibrium policies in general time-inconsistent control problems. Utilizing the extended Hamilton-Jacobi-Bellman system, we recast the original time-inconsistent problem into an equivalent two-stage problem. In the first stage, for given auxiliary functions, we employ the deterministic policy gradient approach to learn an optimal policy in an auxiliary time-consistent control problem. In the second stage, given the updated policy, we exploit the inner fixed point iterations and some martingale characterizations to learn the auxiliary functions. As a theoretical contribution, we provide some mild model assumptions and establish the convergence of inner fixed point iterations. By repeating this actor-critic style of iterations across two stages, our algorithm aims to learn the equilibrium under different sources of time-inconsistency in a unified manner. The superior effectiveness of the proposed algorithm are illustrated in two classical financial applications with time-inconsistency: mean-variance portfolio management and optimal tracking portfolio under non-exponential discounting.

A new Block Coordinate Gradient (BCG) method, dubbed bAdag, for smooth, nonconvex minimization problem is proposed; it falls in the class of Objective Function Free Optimization (OFFO) methods, and it is based on the AdaGrad algorithm. At each iteration, our method computes an adaptive step size based on the cumulative sum of block gradients, instead of full gradients as in AdaGrad-type methods. We prove ergodic, sublinear convergence rates for the bAdag algorithm when minimizing a smooth, possibly nonconvex objective under the (block) Lipschitz continuity assumption on the gradient. Our theory covers three widely popular block selection strategies: the Cyclic (C) rule, Uniform Random selection (UR), and the greedy Gauss-Southwell (GS) rule. We also extend our algorithm and its convergence theory to box-constrained smooth functions. We validate the proposed algorithms through synthetic and real-world experiments.

Optimistic Gradient Descent Ascent (OGDA) and Optimistic Multiplicative-Weights Update (OMWU) are two very popular algorithms to solve convex/concave saddle-point problems, where OMWU is the non-Euclidean, entropic version of OGDA. It is known since the '80s that the last iterate of OGDA asymptotically converges to a saddle point in smooth problems. On the other hand, it is unknown if OMWU has the same property. In this paper, I show that OMWU converges asymptotically for smooth convex-concave saddle-point problems, with a small enough constant learning rate. The result does not require uniqueness, strict complementarity, an error bound, or initialization near a solution. The main new ingredient is a boundary argument showing that every cluster point satisfies the inactive-coordinate KKT inequalities. The boundary argument was discovered with assistance from ChatGPT and is documented in the appendix.

Why do similar funding shocks generate sharply different credit outcomes across countries? We develop and estimate a dynamic structural model in which intermediary credit capacity governs the transmission of funding disruptions to lending. Using supervisory data on U.S. banks and credit unions and Brazilian banks and cooperatives from 2002--2025, we recover institution-level credit capacity and its dynamics across major crisis episodes. Credit capacity is three to six times larger in the United States than in Brazil, while persistence is similar across countries. As a result, funding shocks generate substantially larger and more persistent lending contractions in Brazil. Counterfactual analysis shows that differences in baseline credit capacity, rather than persistence, account for most cross-country variation in crisis propagation and policy effectiveness.

In this paper, we develop a generalized differentiation theory at infinity for convex sets and functions. In particular, we study several fundamental notions of convex analysis at infinity, including tangent cones, normal cones, and subdifferentials. Our work complements the recently developed theories of nonsmooth analysis at infinity by focusing on the convex setting, where these constructions preserve convexity and admit natural connections with recession analysis, polarity, and epigraphical geometry. We show that, in the convex case, many results can be established under weaker assumptions and admit simpler and more explicit representations than those available in the general nonsmooth framework. In addition, we develop calculus rules and geometric characterizations for these constructions and apply them to optimality conditions and attainment criteria for convex optimization problems over unbounded feasible sets. The results obtained in this paper provide new tools for the study of convex sets and functions at infinity and further strengthen the connections between convex analysis, variational analysis, and optimization.

We present a novel method for solving conditional value-at-risk (CVaR) optimization problems based on the dual representation of CVaR, which is defined as the worst-case expectation over a risk envelope. The method is based on the Bregman proximal point algorithm and alternates between stochastic primal and dual stages. Every (inner) primal stage involves a subproblem solved by sampling from a probability distribution updated at each dual stage (outer iteration). The likelihood ratio of the dual probability distributions relative to the distribution underlying the original problem converges to the risk identifier of the solution's CVaR. Thus, the dual distribution provides the algorithm with a built-in importance sampling mechanism that draws from the tail of the underlying distribution. Because only samples in the tail influence the CVaR, and samples outside the tail are drawn with decreasing probability, the algorithm delivers exceptional performance over other stochastic approximation methods. We prove the convergence of the algorithm for convex objective functions. Our numerical experiments target representative problems in financial mathematics and machine learning, focusing on portfolio optimization and support-vector machines, respectively.

We investigate the Schrödinger bridge problem on a finite state space with a strictly positive Markov reference kernel. Starting from the semi-dual convex formulation, we introduce a continuous-time evolution for the terminal Schrödinger potential and show that its equilibria coincide with the unique solution of the bridge problem. The proposed dynamics induces an evolution for the terminal marginal. This marginal equation is governed by a state-dependent nonlocal Onsager operator, identified with the Hessian of the semi-dual functional. We derive its associated Dirichlet form, establish coercivity estimates on the appropriate quotient space, and interpret the resulting equation as a nonlocal gradient-flow formulation of relative entropy. Under natural positivity assumptions, we prove global well-posedness of the SBOF, convergence to the Schrödinger bridge, convergence of the induced couplings and path measures, and exponential relaxation of the terminal marginal. The latter follows from a uniform Poincaré inequality on compact sublevel sets together with entropy--variance comparison estimates. We also discuss the connection with finite-state generative modeling through the Doob transform and illustrate the theory on finite-grid examples involving rare states.

We study the problem of fair division of indivisible goods with constraints imposed by a prescribed independence structure, that is, a family of subsets of goods closed under taking subsets. As a motivating example, imagine that the goods to be divided are the available connections in a logistic, financial, or social network. The admissible bundle of goods for each agent must correspond to an acyclic set of edges, corresponding to a basic feasible solution to a linear network problem to be solved. Suppose that all agents assign the same value to each good (in the example, the network connections are equally important for every agent) and evaluate each bundle by summing the values of its goods. Is there a fair partition of the goods into such acyclic bundles? Surprisingly, the answer is yes, provided that the number of agents is at least the arboricity of $G$, and the fairness requirement is envy-freeness up to one good (EF1). The situation becomes more mysterious when agents have arbitrary additive valuations. Our main result guarantees that, in this case, epistemic EF1 partitions always exist, which means that each agent receives an acyclic bundle for which there exists a feasible partition of the remaining goods into acyclic bundles that they do not envy up to one good. We derive this conclusion from a general result for abstract independence structures defined on the sets of goods. We also discuss connections with several conjectures concerning matroids. In particular, we prove that any Hamiltonian matroid partitionable into two independent sets admits an EF1 bipartition with respect to a common monotone valuation. We complement our results with a constructive perspective: we present explicitly two algorithms for computing the fair allocations described above. Finally, we provide illustrative examples to demonstrate these algorithms on specific instances.

We study a mean field game (MFG) of state and control with state dynamics described by stochastic differential equations driven by both idiosyncratic and common noise, and subject to the constraint that the terminal state variable belongs to a nonempty, convex, closed set. The mean-field interaction enters both the dynamics and the costs through state and control. We formulate an equivalent auxiliary MFG problem and derive the stochastic maximum principle for an auxiliary optimization problem under fixed flows. By means of a suitable forward-backward stochastic differential equation (FBSDE) of conditional McKean-Vlasov type, we establish a characterization of MFG solutions via such a system. Moreover, we prove the existence and uniqueness of solutions for the FBSDE system in the specific case where the running cost is zero and the state coefficients are linear. Additionally, we obtain uniqueness results in the sense of Lasry-Lions and we apply our findings to MFGs of optimal investment with a quadratic relative performance criterion.

Let $\Delta$ be the Neumann Laplacian on the interval $(0,\pi)$, and let $T>0$. An integration by parts formula is proven for the spectral fractional Laplacian, $(-\Delta)^\alpha$, for $\alpha \in (0,1)$. As an application, we prove well-posedness results for the structurally damped beam equation $$u_{tt}+\Delta^2 u+\rho (-\Delta)^\alpha u_t=0, x\in (0,\pi),t>0$$ with various boundary conditions including $$ u_x(0,t)=u_{xxx}(0,t)=0;\ u_x(\pi,t)=f(t),\ u_{xxx}(\pi,t)=0, $$ and $f\in L^2(0,T)$ and appropriate initial conditions. Viewing $f$ as a control, we prove null-controllability. Analagous results are proven for higher order controls, and for the Dirichlet Laplacian.

While long treated as empirical fits, Hill functions have been postulated to be the universal Hopfield barrier for sharpness of input-output responses by Martinez-Corral, Nam, DePace, and Gunawardena. A Hopfield barrier is a fundamental limit on how well biological systems can process information without expending energy. Their case rested on numerical findings for Hill coefficients $4$ and $6$. We give a precise formulation and proof of this: measuring sharpness by the supremum of the derivative in semi-log scale, any rational function $r(x)=(\alpha_0+\alpha_1 x+ \cdots +\alpha_n x^n)/(\beta_0 + \beta_1 x+ \cdots + \beta_n x^n)$ with real coefficients $0\leq \alpha_i\leq \beta_i$ has sharpness at most $n/4$, with equality if and only if $r$ is a Hill function with Hill coefficient $n$.

We propose Annealed Entropic Allocation, an annealed weighted soft-min framework for sequential budget allocation in ranking and selection. The central idea is to replace the non-smooth maximin large-deviation rate objective with a weighted log-sum-exp surrogate that aggregates challenger-specific pairwise scores through soft-min weights, mitigating hard switching when several challengers are nearly active. To improve finite-budget discrimination, we incorporate the saddlepoint approximation -- a sub-exponential correction derived from refined pairwise tail asymptotics. Because these corrections are sub-exponential and the smoothing parameter is annealed to zero, the surrogate preserves the same first-order large-deviation target as the classical maximin formulation. We show that the surrogate converges uniformly to the hard minimum, that the soft-min weights concentrate on the active challengers, and that, under fixed weights, the induced target allocation map is continuous on the simplex interior. Numerical experiments on Gaussian and exponential instances demonstrate competitive performance, especially when multiple challengers are nearly tied.