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This paper introduces a novel control strategy for agile spacecraft attitude control, addressing reaction wheel-related input and state constraints. An optimal-decay control Lyapunov function quadratic program stabilizes the system and mitigates chattering at low sampling frequencies, while control barrier functions enforce hard state constraints. Numerical simulations validate the method's practicality and efficiency for real-time agile spacecraft attitude control.

Backtracking line search is foundational in numerical optimization. The basic idea is to adjust the step size of an algorithm by a constant factor until some chosen criterion (e.g. Armijo, Goldstein, Descent Lemma) is satisfied. We propose a new way for adjusting step sizes, replacing the constant factor used in regular backtracking with one that takes into account the degree to which the chosen criterion is violated, without additional computational burden. For convex problems, we prove adaptive backtracking requires fewer adjustments to produce a feasible step size than regular backtracking does for two popular line search criteria: the Armijo condition and the descent lemma. For nonconvex smooth problems, we additionally prove adaptive backtracking enjoys the same guarantees of regular backtracking. Finally, we perform a variety of experiments on over fifteen real world datasets, all of which confirm that adaptive backtracking often leads to significantly faster optimization.

For the three body problem with equal masses, we prove that the most symmetric continuation class of Lagrange's equilateral triangle solution, also referred to as the $P_{12}$ family of Marchal, contains the remarkable figure eight choreography discovered by Moore in 1993, and proven to exist by Chenciner and Montgomery in 2000. This settles a conjecture of Marchal which dates back to the 1999 conference on Celestial Mechanics in Evanston Illinois, celebrating Donald Saari's 60th birthday.

This vision paper focuses on the mental health crisis impacting healthcare workers (HCWs), which exacerbated by the COVID-19 pandemic, leads to increased stress and psychological issues like burnout. Peer Support Programs (PSP) are a recognized intervention for mitigating these issues. These programs are increasingly being delivered virtually through Peer Support Systems (PSS) for increased convenience and accessibility. However, HCWs perception of these systems results in fear of information sharing, perceived lack of safety, and low participation rate, which challenges these systems ability to achieve their goals. In line with the rich body of research on the requirements and properties of trustworthy systems, we posit that increasing HCWs trust in PSS could address these challenges. However, extant research focuses on objectively defined trustworthiness rather than perceptual trust because trustworthy requirements are viewed as more controllable and easier to operationalize. This study proposes a novel approach to elicit perceptual trust requirements by proposing a trust framework anchored in recognized trust theories from different disciplines that unpacks trust into its recognized types and their antecedents. This approach allows the identification of trust requirements beyond those already proposed for trustworthy systems, providing a strong foundation for improving the effectiveness of PSS for HCWs. Keywords: Trust Requirements, Requirements elicitation, Peer support systems, Healthcare workers, Comment: Accepted for publication at the RE@Next! track of RE 2024

In the dynamics generated by the suspension bridge equation, traveling waves are an essential feature. The existing literature focuses primarily on the idealized one-dimensional case, while traveling structures in two spatial dimensions have only been studied via numerical simulations. We use computer-assisted proof methods based on a Newton-Kantorovich type argument to find and prove periodic localized traveling waves in two dimensions. The main obstacle is the exponential nonlinearity in combination with the resulting large amplitude of the localized waves. Our analysis hinges on establishing computable bounds to control the aliasing error in the computed Fourier coefficients. This leads to existence proofs of different traveling wave solutions, accompanied by small, explicit, rigorous bounds on the deficiency of numerical approximations. This approach is directly extendable to other wave equation models and elliptic partial differential equations with analytic nonlinearities, in two as well as in higher dimensions., Comment: Added a reference

This paper introduces a novel computer-assisted method for detecting and constructively proving the existence of cusp bifurcations in differential equations. The approach begins with a two-parameter continuation along which a tool based on the theory of Poincar\'e index is employed to identify the presence of a cusp bifurcation. Using the approximate cusp location, Newton's method is then applied to a given augmented system (the cusp map), yielding a more precise numerical approximation of the cusp. Through a successful application of a Newton-Kantorovich type theorem, we establish the existence of a non-degenerate zero of the cusp map in the vicinity of the numerical approximation. Employing a Gershgorin circles argument, we then prove that exactly one eigenvalue of the Jacobian matrix at the cusp candidate has zero real part, thus rigorously confirming the presence of a cusp bifurcation. Finally, by incorporating explicit control over the cusp's location, a rigorous enclosure for the normal form coefficient is obtained, providing the explicit dynamics on the center manifold at the cusp. We show the effectiveness of this method by applying it to four distinct models.

In this paper, we present a methodology for establishing constructive proofs of existence of smooth, stationary, non-radial localized patterns in the planar Swift-Hohenberg equation. Specifically, given an approximate solution $u_0$, we construct an approximate inverse for the linearization around $u_0$, enabling the development of a Newton-Kantorovich approach. Consequently, we derive a sufficient condition for the existence of a unique localized pattern in the vicinity of $u_0$. The verification of this condition is facilitated through a combination of analytic techniques and rigorous numerical computations. Moreover, an additional condition is derived, establishing that the localized pattern serves as the limit of a family of periodic solutions (in space) as the period tends to infinity. The integration of analytical tools and meticulous numerical analysis ensures a comprehensive validation of this condition. To illustrate the efficacy of the proposed methodology, we present computer-assisted proofs for the existence of three distinct unbounded branches of periodic solutions in the planar Swift-Hohenberg equation, all converging towards a localized planar pattern, whose existence is also proven constructively. All computer-assisted proofs, including the requisite codes, are accessible on GitHub at \cite{julia_cadiot}.

We present our position on the elusive quest for a general-purpose framework for specifying and studying deep learning architectures. Our opinion is that the key attempts made so far lack a coherent bridge between specifying constraints which models must satisfy and specifying their implementations. Focusing on building a such a bridge, we propose to apply category theory -- precisely, the universal algebra of monads valued in a 2-category of parametric maps -- as a single theory elegantly subsuming both of these flavours of neural network design. To defend our position, we show how this theory recovers constraints induced by geometric deep learning, as well as implementations of many architectures drawn from the diverse landscape of neural networks, such as RNNs. We also illustrate how the theory naturally encodes many standard constructs in computer science and automata theory., Comment: To appear in ICML 2024. Comments welcome. More info at categoricaldeeplearning.com

In this paper, we introduce a general constructive method to compute solutions of initial value problems of semilinear parabolic partial differential equations via semigroup theory and computer-assisted proofs. Once a numerical candidate for the solution is obtained via a finite dimensional projection, Chebyshev series expansions are used to solve the linearized equations about the approximation from which a solution map operator is constructed. Using the solution operator (which exists from semigroup theory), we define an infinite dimensional contraction operator whose unique fixed point together with its rigorous bounds provide the local inclusion of the solution. Applying this technique for multiple time steps leads to constructive proofs of existence of solutions over long time intervals. As applications, we study the 3D/2D Swift-Hohenberg, where we combine our method with explicit constructions of trapping regions to prove global existence of solutions of initial value problems converging asymptotically to nontrivial equilibria. A second application consists of the 2D Ohta-Kawasaki equation, providing a framework for handling derivatives in nonlinear terms.

In this paper, we study a finite population undergoing discrete, nonoverlapping generations, that is structured into $D$ demes, each containing $N$ individuals of two possible types, $A$ and $B$, whose viability coefficients, $s_A$ and $s_B$, respectively, vary randomly from one generation to the next. We assume that the means, variances and covariance of the viability coefficients are inversely proportional to the number of demes $D$, while higher-order moments are negligible in comparison to $1/D$. We use a discrete-time Markov chain with two time scales to model the evolutionary process, and we demonstrate that as the number of demes $D$ approaches infinity, the accelerated Markov chain converges to a diffusion process for any deme size $N\geq 2$. This diffusion process allows us to evaluate the fixation probability of type $A$ following its introduction as a single mutant in a population that was fixed for type $B$. We explore the impact of increasing the variability in the viability coefficients on this fixation probability. At least when $N$ is large enough, it is shown that increasing this variability for type $B$ or decreasing it for type $A$ leads to an increase in the fixation probability of a single $A$. The effect of the population-scaled variances, $\sigma^2_A$ and $\sigma^2_B$, can even cancel the effects of the population-scaled means, $\mu_A$ and $\mu_B$. We also show that the fixation probability of a single $A$ increases as the deme-scaled migration rate increases. Moreover, this probability is higher for type $A$ than for type $B$ if the population-scaled geometric mean is higher for type $A$ than for type $B$, which means that $\mu_A-\sigma_A^2/2>\mu_B-\sigma_B^2/2$.

In this paper, we consider a two-player two-strategy game with random payoffs in a population subdivided into $d$ demes, each containing $N$ individuals at the beginning of any given generation and experiencing local extinction and recolonization with some fixed probability $m$ after reproduction and selection among offspring. Within each deme, offspring engage in random pairwise interactions, and the payoffs are assumed to have means and variances proportional to the inverse of the population size. By verifying the conditions given in Ethier and Nagylaki (1980) to approximate Markov chains with two time scales, we establish that the discrete-time evolutionary dynamics with $Nd$ generations as unit of time converges to a continuous-time diffusion as $d\rightarrow\infty$. The infinitesimal mean and variance of this diffusion are expressed in terms of the population-scaled means and variances of the payoffs besides identity-by-descent measures between offspring in the same deme in a neutral population. We show that the probability for a strategy to fix in the population starting from an initial frequency $(Nd)^{-1}$ generally increases as the payoffs to that strategy exhibit less variability or the payoffs to the other strategy more variability. As a result, differences in variability can make this fixation probability for cooperation larger than the corresponding one for defection. As the deme-scaled extinction rate $\nu=mN$ decreases for $N$ large enough and $m$ small enough, creating a higher level of identity among offspring within demes, the differences between the population-scaled variances of the payoffs for interacting offspring of different types increases this effect to a greater extent than the differences for interacting offspring of the same type.

In order to better understand the impact of environmental stochastic fluctuations on the evolution of animal behavior, we introduce the concept of a stochastic Nash equilibrium (SNE) that extends the classical concept of a Nash equilibrium (NE). Based on a stochastic stability analysis of a linear evolutionary game with temporally varying payoffs, we address the question of the existence of a SNE, either weak when the geometric mean payoff against it is the same for all other strategies or strong when it is strictly smaller for all other strategies, and its relationship with a stochastically evolutionarily stable (SES) strategy. While a strong SNE is always SES, this is not necessarily the case for a weak SNE. We give conditions for a completely mixed weak SNE not to be SES and to coexist with at least two strong SNE. More importantly, we show that a pair of two completely mixed strong SNE can emerge as the noise level increases. This not only indicates that a noise-induced SNE may possess some properties that a NE cannot possess, such as being completely mixed and strong, but also illustrates the complexity of evolutionary game dynamics in a stochastic environment.

We present Model Predictive Planning (MPP), a trajectory planner for low-agility vehicles such as a fixed-wing aircraft to navigate obstacle-laden environments. MPP consists of (1) a multi-path planning procedure that identifies candidate paths, (2) a raytracing procedure that generates linear constraints around these paths to enforce obstacle avoidance, and (3) a convex quadratic program that finds a feasible trajectory within these constraints if one exists. Low-agility aircraft cannot track arbitrary paths, so refining a given path into a trajectory that respects the vehicle's limited maneuverability and avoids obstacles often leads to an infeasible optimization problem. The critical feature of MPP is that it efficiently considers multiple candidate paths during the refinement process, thereby greatly increasing the chance of finding a feasible and trackable trajectory. We demonstrate the effectiveness of MPP on a longitudinal aircraft model.

We consider the distributed optimization problem for a multi-agent system. Here, multiple agents cooperatively optimize an objective by sharing information through a communication network and performing computations. In this tutorial, we provide an overview of the problem, describe the structure of its algorithms, and use simulations to illustrate some algorithmic properties based on this structure., Comment: 6 pages, 14 figures, to appear at IEEE Conference on Decision and Control 2023

Iterative gradient-based optimization algorithms are widely used to solve difficult or large-scale optimization problems. There are many algorithms to choose from, such as gradient descent and its accelerated variants such as Polyak's Heavy Ball method or Nesterov's Fast Gradient method. It has long been observed that iterative algorithms can be viewed as dynamical systems, and more recently, as robust controllers. Here, the "uncertainty" in the dynamics is the gradient of the function being optimized. Therefore, worst-case or average-case performance can be analyzed using tools from robust control theory, such as integral quadratic constraints (IQCs). In this tutorial paper, we show how such an analysis can be carried out using an alternative Lyapunov-based approach. This approach recovers the same performance bounds as with IQCs, but with the added benefit of constructing a Lyapunov function., Comment: 6 pages, 3 figures, to appear at IEEE Conference on Decision and Control 2023

Primal-dual algorithms are frequently used for iteratively solving large-scale convex optimization problems. The analysis of such algorithms is usually done on a case-by-case basis, and the resulting guaranteed rates of convergence can be conservative. Here we consider a class of first-order algorithms for linearly constrained convex optimization problems, and provide a linear matrix inequality (LMI) analysis framework for certifying worst-case exponential convergence rates. Our approach builds on recent results for interpolation of convex functions and linear operators, and our LMI directly constructs a Lyapunov function certifying the guaranteed convergence rate. By comparing to rates established in the literature, we show that our approach can certify significantly faster convergence for this family of algorithms., Comment: 6 pages, 2 figures, to appear at IEEE Conference on Decision and Control 2023

Hyperspectral unmixing aims at decomposing a given signal into its spectral signatures and its associated fractional abundances. To improve the accuracy of this decomposition, algorithms have included different assumptions depending on the application. The goal of this study is to develop a new unmixing algorithm that can be applied for the calibration of multi-point scintillation dosimeters used in the field of radiation therapy. This new algorithm is based on a non-negative matrix factorization. It incorporates a partial prior knowledge on both the abundances and the endmembers of a given signal. It is shown herein that, following a precise calibration routine, it is possible to use partial prior information about the fractional abundances, as well as on the endmembers, in order to perform a simplified yet precise calibration of these dosimeters. Validation and characterization of this algorithm is made using both simulations and experiments. The experimental validation shows an improvement in accuracy compared to previous algorithms with a mean spectral angle distance (SAD) on the estimated endmembers of 0.0766, leading to an average error of $(0.25 \pm 0.73)$ % on dose measurements., Comment: 12 pages, 6 figures

We consider the $N$-vortex problem on the sphere assuming that all vorticities have equal strength. We investigate relative equilibria (RE) consisting of $n$ latitudinal rings which are uniformly rotating about the vertical axis with angular velocity $\omega$. Each such ring contains $m$ vortices placed at the vertices of a concentric regular polygon and we allow the presence of additional vortices at the poles. We develop a framework to prove existence and orbital stability of branches of RE of this type parametrised by $\omega$. Such framework is implemented to rigorously determine and prove stability of segments of branches using computer-assisted proofs. This approach circumvents the analytical complexities that arise when the number of rings $n\geq 2$ and allows us to give several new rigorous results. We exemplify our method providing new contributions consisting in the determination of enclosures and proofs of stability of several equilibria and RE for $5\leq N\leq 12$., Comment: 60 pages. Second version includes a Data Accessibility Statement and a link to a GitHub repository where the code to reproduce the computer assisted proofs is available

We raise concerns about controllers' robustness in simple reinforcement learning benchmark problems. We focus on neural network controllers and their low neuron and symbolic abstractions. A typical controller reaching high mean return values still generates an abundance of persistent low-return solutions, which is a highly undesirable property, easily exploitable by an adversary. We find that the simpler controllers admit more persistent bad solutions. We provide an algorithm for a systematic robustness study and prove existence of persistent solutions and, in some cases, periodic orbits, using a computer-assisted proof methodology., Comment: accepted to ECAI23

It remains to be ascertained whether sub-Neptune exoplanets primarily possess hydrogen-rich atmospheres or whether a population of H$_2$O-rich "water worlds" lurks in their midst. Addressing this question requires improved modeling of water-rich exoplanetary atmospheres, both to predict and interpret spectroscopic observations and to serve as upper boundary conditions on interior structure calculations. Here we present new models of hydrogen-helium-water atmospheres with water abundances ranging from solar to 100% water vapor. We improve upon previous models of high water content atmospheres by incorporating updated prescriptions for water self-broadening and a non-ideal gas equation of state. Our model grid (https://umd.box.com/v/water-worlds) includes temperature-pressure profiles in radiative-convective equilibrium, along with their associated transmission and thermal emission spectra. We find that our model updates primarily act at high pressures, significantly impacting bottom-of-atmosphere temperatures, with implications for the accuracy of interior structure calculations. Upper atmosphere conditions and spectroscopic observables are less impacted by our model updates, and we find that under most conditions, retrieval codes built for hot Jupiters should also perform well on water-rich planets. We additionally quantify the observational degeneracies among both thermal emission and transmission spectra. We recover standard degeneracies with clouds and mean molecular weight for transmission spectra, and we find thermal emission spectra to be more readily distinguishable from one another in the water-poor (i.e. near-solar) regime., Comment: Accepted for publication in ApJ. Full model grid is available at https://umd.box.com/v/water-worlds

Recently, the first JWST measurement of thermal emission from a rocky exoplanet was reported. The inferred dayside brightness temperature of TRAPPIST-1 b at 15 $\mu$m is consistent with the planet having no atmosphere and therefore no mechanism by which to circulate heat to its nightside. In this Letter, we compare the measured secondary eclipse depth of TRAPPIST-1 b to predictions from a suite of self-consistent radiative-convective equilibrium models in order to quantify the maximum atmospheric thickness consistent with the observation. We find that plausible atmospheres (i.e., those that contain at least 100 ppm CO$_2$) with surface pressures greater than 0.01 bar (0.1 bar) are ruled out at 1$\sigma$ (3$\sigma$), regardless of the choice of background atmosphere. Thicker atmospheres of up to 10 bar (100 bar) at 1$\sigma$ (3$\sigma$) are only allowed if the atmosphere lacks any strong absorbers across the mid-IR wavelength range, a scenario that we deem unlikely. We additionally model the emission spectra for bare-rock planets of various compositions. We find that a variety of silicate surfaces match the measured eclipse depth to within 1$\sigma$, and the best-fit grey albedo is $0.02 \pm 0.11$. We conclude that planned secondary eclipse observations at 12.8 $\mu$m will serve to validate the high observed brightness temperature of TRAPPIST-1 b, but are unlikely to further distinguish among the consistent atmospheric and bare-rock scenarios., Comment: 8 pages, 4 figures, submitted to ApJL

It is well-known that linear quadratic regulators (LQR) enjoy guaranteed stability margins, whereas linear quadratic Gaussian regulators (LQG) do not. In this letter, we consider systems and compensators defined over directed acyclic graphs. In particular, there are multiple decision-makers, each with access to a different part of the global state. In this setting, the optimal LQR compensator is dynamic, similar to classical LQG. We show that when sub-controller input costs are decoupled (but there is possible coupling between sub-controller state costs), the decentralized LQR compensator enjoys similar guaranteed stability margins to classical LQR. However, these guarantees disappear when cost coupling is introduced.

We present a computational method for studying transverse homoclinic orbits for periodic solutions of delay differential equations, a phenomenon that we refer to as the \emph{Poincar\'{e} scenario}. The strategy is geometric in nature, and consists of viewing the connection as the zero of a nonlinear map, such that the invertibility of its Fr\'{e}chet derivative implies the transversality of the intersection. The map is defined by a projected boundary value problem (BVP), with boundary conditions in the (finite dimensional) unstable and (infinite dimensional) stable manifolds of the periodic orbit. The parameterization method is used to compute the unstable manifold and the BVP is solved using a discrete time dynamical system approach (defined via the \emph{method of steps}) and Chebyshev series expansions. We illustrate this technique by computing transverse homoclinic orbits in the cubic Ikeda and Mackey-Glass systems.