arXiv subjects

Eduardo D. Sontag

Publications and source records attributed to Eduardo D. Sontag.

At least 19 recordsLinked to original sources

arXiv2026

On non-monotonic steady state response curves

Non-monotonic steady state response curves can readily arise from an incoherent feedforward loop (IFFL), which has also been conjectured to be a necessary mechanism. In a previous work, we showed this conjecture to be false for systems where the control parameter directly influences only one state variable. Here, we extend this result to arbitrary systems. We show that a stable non-monotonic response requires either an IFFL or a combination of positive and negative feedback loops. Finally, we apply our theoretical findings to three network-structured biological contexts: signaling pathways, gene regulation, and ecological consumer-resource models. For the first two contexts, we identify necessary motifs for non-monotonic responses, while for consumer-resource models, we provide an alternative proof of the Competitive Exclusion Principle.

q-bio.QM↗

arXiv2026

Discrete-time polynomial systems with inputs and outputs: structure, reachability, observability, and minimal realizations

We study the realization problem for discrete-time input/output polynomial systems. These are formalized using tools from commutative algebra and algebraic geometry as systems whose state spaces are algebraic varieties, or more abstractly the set of k-points of an affine k-scheme, where k is an arbitrary infinite field. The input/output behaviors of such systems are described by "polynomial response maps" in which outputs are polynomial functions of past inputs. The main results show that every polynomial response map admits a canonical (quasi-reachable and algebraically observable) realization, which is unique up to isomorphism. The key to the approach is to linearize dynamics by considering a "dual" system in which states are functions defined on states. Finite dimensionality of the canonical realization, and its polynomiality, are characterized in terms of the space, algebra, and field of observables of the map, as well as in terms of algebraic input/output difference equations, and by a Jacobian rank criterion. A particular subclass consists of the maps that we call "bounded," defined by the property that their degree in the past inputs is uniformly bounded. Bounded maps are shown to be finitely realizable if and only if they are realizable by finite-dimensional state-affine systems, whose theory in turn reduces to that of rational formal power series. We also study the lattice of quasi-reachable realizations of a given map, including normal realizations. This work is an update of the PhD thesis written by the author in 1976; connections to recent work are briefly discussed in the last section.

eess.SY↗

arXiv2026

A note on bistability of a two-gene competitive system

Self-regulation together with mutual promoter competition provides a simple mechanism for bistability in gene-regulatory models. We study a two-gene system with regulatory terms of Hill exponent one, allowing distinct basal production rates and distinct degradation rates. Each gene product, when bound to its own promoter, may enhance or reduce production relative to the basal rate, while the two products compete through promoter occupancy. We show that the system has at least one and at most three equilibria in the positive quadrant. Exactly two positive equilibria can occur only if one nullcline intersection is degenerate; consequently, a configuration in which all positive nullcline intersections are transverse has either one or three positive equilibria. If there are exactly three distinct positive equilibria, then all three are automatically hyperbolic: the two outer equilibria are asymptotically stable nodes and the middle equilibrium is a saddle. Moreover, every positive solution converges to an equilibrium. Hence the positive quadrant is the disjoint union of the basins of attraction of the two stable nodes and the one-dimensional stable manifold of the saddle, yielding global bistability.

math.DS↗

arXiv2026

Convergence guarantees for Muon: New parameter regimes and generalizations

In this paper, we establish the first asymptotic convergence guarantees for the Muon algorithm through a more accurate proxy for the Newton-Schultz iteration than the typical matrix sign function. We prove that, for appropriate choices of hyperparameters, the iterates satisfy $\lim_{k\to\infty}\|\nabla f(x_k)\|=0$, and, under a global Polyak-Łojasiewicz condition, that the sequence of function values converges linearly. The key insight is that the regularization, implicit in Muon's Newton-Schulz implementation, induces a bounded preconditioner, exposing Muon as a \emph{preconditioned Polyak heavy-ball} method and enabling a classical Lyapunov analysis. This observation naturally motivates applying the same preconditioning structure to the Nesterov gradient evaluation. We formalize this idea by introducing \emph{Muesterov}, a Nesterov-based variant of Muon, and prove that it enjoys the same convergence guarantees, extending the theoretical framework beyond the heavy-ball setting. Numerical experiments on a scalar cross-entropy problem corroborate the theory and illuminate the joint role of the learning rate and the Newton-Schulz regularizer in controlling convergence. Preliminary numerical simulations training the nanoGPT dataset provide intuition regarding the relevance of the observations in this paper to practical applications.

math.NA↗

arXiv2026

Dynamic response phenotypes and model discrimination in systems and synthetic biology

Biological systems encode function not primarily in steady states, but in the structure of transient responses elicited by time-varying stimuli. Overshoots, biphasic dynamics, adaptation kinetics, fold-change detection, entrainment, and cumulative exposure effects often determine phenotypic outcomes, yet are poorly captured by classical steady-state or dose-response analyses. This paper develops an input-output perspective on such "dynamic phenotypes," emphasizing how qualitative features of transient behavior constrain underlying network architectures independently of detailed parameter values. A central theme is the role of sign structure and interconnection logic, particularly the contrast between monotone systems and architectures containing antagonistic pathways. We show how incoherent feedforward (IFF) motifs provide a simple and recurrent mechanism for generating non-monotonic and adaptive responses across multiple levels of biological organization, from molecular signaling to immune regulation and population dynamics. Conversely, monotonicity imposes sharp impossibility results that can be used to falsify entire classes of models from transient data alone. Beyond step inputs, we highlight how periodic forcing, ramps, and integral-type readouts such as cumulative dose responses offer powerful experimental probes that reveal otherwise hidden structure, separate competing motifs, and expose invariances such as fold-change detection. Throughout, we illustrate how control-theoretic concepts, including monotonicity, equivariance, and input-output analysis, can be used not as engineering metaphors, but as precise mathematical tools for biological model discrimination. Thus we argue for a shift in emphasis from asymptotic behavior to transient and input-driven dynamics as a primary lens for understanding, testing, and reverse-engineering biological networks.

q-bio.QM↗

arXiv2026

Smooth globally PLI functions are nonlinear least-squares, and so are their gradient-dominated cousins

Boumal, Criscitiello and Rebjock (BCR) proved that if $M$ is a contractible, connected and complete Riemannian manifold, then every smooth function $f\colon M\to R$ satisfying the global Polyak--Łojasiewicz inequality (PŁI) is necessarily of the form $f = f^* + \|ϕ\|^2$ with $ϕ$ a submersion. Informally, minimizing such a function amounts to solving a nonlinear least-squares problem in new coordinates. The global PŁI hypothesis fails, however, in many problems of interest, among them continuous-time LQR policy optimization in optimal control and a standard formulation of logistic regression. A hierarchy of weakened PŁ inequalities has been introduced in order to cover such problems, and more generally to study the effect of noise and adversarial perturbations on gradient flows. This note shows that, with minor modifications, the same reduction to a nonlinear least-squares problem holds under a substantially weaker hypothesis, ``semiglobal'' PŁI, which is satisfied in both of the examples just mentioned. That condition asks that $f$ satisfy an estimate $\|\nabla f(x)\| \ge α\bigl(f(x)-f^*\bigr)$ for all $x$, with $α$ merely positive definite and bounded below by a positive multiple of $\sqrt{s}$ for small $s>0$.

eess.SY↗

arXiv2026

On input-output persistency and the interconnection of positive nonlinear systems

In this paper we study feedback interconnections of positive nonlinear SISO systems and introduce two complementary properties: persistent-input/persistent-output (PIPO) for the plant and persistent-input/transient-output (PITO) for the controller. Assuming forward completeness of the closed loop and a one-sided affine growth bound on the controller output, we show that PIPO and PITO jointly imply boundedness of the control signal. The result is input-output in nature and does not require linearity or monotonicity of the interconnected subsystems, although it does not in general guarantee boundedness of the full controller state. We provide a structural PIPO condition for positive monotone plants with a class-$\mathcal K_\infty$ steady-state characteristic, establish PITO for the antithetic integral controller with explicit gains, and illustrate the framework on a nonlinear integral-feedback motif with multiplicative controller growth.

eess.SY↗

arXiv2026

Some intuition for why cooperative systems "look 1-dimensional" and 2-cooperative systems "look 2-dimensional"

It is known that cooperative systems, and more generally systems monotone with respect to cones, behave under appropriate irreducibility conditions like one-dimensional systems, as evidenced by results such as Hirsch's generic convergence theorem. It is also known, following work of Sanchez and others, that two-cooperative systems, those preserving a generally nonconvex 2-dimensional cone (tested through the diminishing of sign variations), behave like two-dimensional systems, as evidenced by Poincare--Bendixson-type theorems. In these notes I attempt to give some geometric intuition for these dimensionality reductions, based on Birkhoff--Hilbert contractions of the projective metric on a positive cone.

eess.SY↗

arXiv2026

Dynamics and dose response in scaffold ligand binding

This paper considers systems in which two or more ligands bind independently to a common scaffold. Such systems arise in a range of applications, including immunotherapy and synthetic biology. We show that each stoichiometric compatibility class contains a unique steady state, and that this steady state is asymptotically stable. The main result gives a rigorous proof that the steady-state concentration of the fully bound complex, viewed as a function of the total scaffold concentration, has a unique maximum. This biphasic dose response is a characteristic feature of scaffolding systems and, in the special case of two ligands, plays an important role in the design and analysis of bispecific antibody drugs.

physics.bio-ph↗

arXiv2026

Boundedness of solutions in feedback systems with antithetic controllers

Antithetic feedback controllers have become a key experimental and theoretical tool in synthetic biology. Introduced by Khammash and collaborators about 10 years ago, they are employed in order to achieve the practical regulation of protein expression, including tracking and robust disturbance rejection. In closed-loop, there are unique equilibria which, depending on parameter values, can be unstable. It had been shown, however, that this instability is not arbitrary: any bounded trajectory that stays away from the equilibrium must converge to a periodic orbit. This motivated a long-standing open question: is every trajectory bounded? In other words, even if the equilibrium is unstable, can nonlinear effects prevent unbounded excursions in the state space? This paper provides an affirmative answer, establishing the boundedness of all solutions. Previous attempts to prove this fact using Lyapunov functions had no success. Instead, this paper takes a completely different approach, specific to antithetic configurations, in which the key idea is to think of the controller as providing a ``persistently negative feedback'' which acts far away from the equilibrium in such a way so as to keep trajectories from diverging. This new approach, although tailored to the antithetic controller, might be useful in other applications as well.

math.OC↗

arXiv2026

When is cumulative dose response monotonic? Analysis of incoherent feedforward motifs

We study the monotonicity of the cumulative dose response (cDR) for a class of incoherent feedforward motifs (IFFM) systems with linear intermediate dynamics and nonlinear output dynamics. While the instantaneous dose response (DR) may be nonmonotone with respect to the input, the cDR can still be monotone. To analyze this phenomenon, we derive an integral representation of the sensitivity of cDR with respect to the input and establish general sufficient conditions for both monotonicity and non-monotonicity. These results reduce the problem to verifying qualitative sign properties along system trajectories. We apply this framework to four canonical IFFM systems and obtain a complete characterization of their behavior. In particular, IFFM1 and IFFM3 exhibit monotone cDR despite potentially non-monotone DR, while IFFM2 is monotone already at the level of DR, which implies monotonicity of cDR. In contrast, IFFM4 violates these conditions, leading to a loss of monotonicity. Numerical simulations indicate that these properties persist beyond the structured initial conditions used in the analysis. Overall, our results provide a unified framework for understanding how network structure governs monotonicity in cumulative input-output responses.

math.DS↗

arXiv2026

Universal Formulas for Safe Control and Their Neural Network Approximations

We study the problem of designing a controller that satisfies an arbitrary number of affine inequalities at every point in the state space. This is motivated by the fact that a variety of key control objectives, such as stability, safety, and input saturation, are guaranteed by closed-loop systems whose controllers satisfy such inequalities. Many works in the literature design such controllers as the solution to a state-dependent quadratic program (QP) whose constraints are precisely the inequalities. When the input dimension and number of constraints are high, computing a solution of this QP in real time can become computationally burdensome. Additionally, the solution of such optimization problems is not smooth in general, which can degrade the performance of the system. This paper provides a novel method to design a smooth controller that satisfies an arbitrary number of affine constraints. The controller is given at every state as the minimizer of a strictly convex function. To avoid computing the minimizer of such function in real time, we introduce a method based on neural networks (NN) to approximate the controller. Remarkably, this NN can be used to solve the controller design problem for any task with less than a fixed input dimension and number of affine constraints, and is completely independent of the state dimension. This is why we refer to such NN approximation as a NN-based universal formula for control. Additionally, we show that the NN-based controller only needs to be trained with datapoints from a bounded set in the state space, which significantly simplifies the training process.

math.OC↗

arXiv2026

Learning Genetic Circuit Modules with Neural Networks: Full Version

In several applications, including in synthetic biology, one often has input/output data on a system composed of many modules, and although the modules' input/output functions and signals may be unknown, knowledge of the composition architecture can significantly reduce the amount of training data required to learn the system's input/output mapping. Learning the modules' input/output functions is also necessary for designing new systems from different composition architectures. Here, we propose a modular learning framework, which incorporates prior knowledge of the system's compositional structure to (a) identify the composing modules' input/output functions from the system's input/output data and (b) achieve this by using a reduced amount of data compared to what would be required without knowledge of the compositional structure. To achieve this, we introduce the notion of modular identifiability, which allows recovery of modules' input/output functions from a subset of the system's input/output data, and provide theoretical guarantees on a class of systems motivated by genetic circuits. We demonstrate the theory on computational studies showing that a neural network (NNET) that accounts for the compositional structure can learn the composing modules' input/output functions and predict the system's output on inputs outside of the training set distribution. By contrast, a neural network that is agnostic of the structure is unable to predict on inputs that fall outside of the training set distribution. By reducing the need for experimental data and allowing module identification, this framework offers the potential to ease the design of synthetic biological circuits and of multi-module systems more generally.

cs.LG↗

arXiv2026

Remarks on Lipschitz-Minimal Interpolation: Generalization Bounds and Neural Network Implementation

This note establishes a theoretical framework for finding (potentially overparameterized) approximations of a function on a compact set with a-priori bounds for the generalization error. The approximation method considered is to choose, among all functions that (approximately) interpolate a given data set, one with a minimal Lipschitz constant. The paper establishes rigorous generalization bounds over practically relevant classes of approximators, including deep neural networks. It also presents a neural network implementation based on Lipschitz-bounded network layers and an augmented Lagrangian method. The results are illustrated for a problem of learning the dynamics of an input-to-state stable system with certified bounds on simulation error.

eess.SY↗

arXiv2025

Autoencoding Dynamics: Topological Limitations and Capabilities

Given a "data manifold" $M\subset \mathbb{R}^n$ and "latent space" $\mathbb{R}^\ell$, an autoencoder is a pair of continuous maps consisting of an "encoder" $E\colon \mathbb{R}^n\to \mathbb{R}^\ell$ and "decoder" $D\colon \mathbb{R}^\ell\to \mathbb{R}^n$ such that the "round trip" map $D\circ E$ is as close as possible to the identity map $\mbox{id}_M$ on $M$. We present various topological limitations and capabilites inherent to the search for an autoencoder, and describe capabilities for autoencoding dynamical systems having $M$ as an invariant manifold.

cs.LG↗

arXiv2025

On the (almost) Global Exponential Convergence of the Overparameterized Policy Optimization for the LQR Problem

In this work we study the convergence of gradient methods for nonconvex optimization problems -- specifically the effect of the problem formulation to the convergence behavior of the solution of a gradient flow. We show through a simple example that, surprisingly, the gradient flow solution can be exponentially or asymptotically convergent, depending on how the problem is formulated. We then deepen the analysis and show that a policy optimization strategy for the continuous-time linear quadratic regulator (LQR) (which is known to present only asymptotic convergence globally) presents almost global exponential convergence if the problem is overparameterized through a linear feed-forward neural network (LFFNN). We prove this qualitative improvement always happens for a simplified version of the LQR problem and derive explicit convergence rates for the gradient flow. Finally, we show that both the qualitative improvement and the quantitative rate gains persist in the general LQR through numerical simulations.

math.OC↗

arXiv2025

Small-Covariance Noise-to-State Stability of Stochastic Systems and Its Applications to Stochastic Gradient Dynamics

This paper studies gradient dynamics subject to additive random noise, which may arise from sources such as stochastic gradient estimation, measurement noise, or stochastic sampling errors. To analyze the robustness of such stochastic gradient systems, the concept of small-covariance noise-to-state stability (NSS) is introduced, along with a Lyapunov-based characterization. Furthermore, the classical Polyak-Lojasiewicz (PL) condition on the objective function is generalized to the $\mathcal{K}$-PL condition via comparison functions, thereby extending its applicability to a broader class of optimization problems. It is shown that the stochastic gradient dynamics exhibit small-covariance NSS if the objective function satisfies the $\mathcal{K}$-PL condition and possesses a globally Lipschitz continuous gradient. This result implies that the trajectories of stochastic gradient dynamics converge to a neighborhood of the optimum with high probability, with the size of the neighborhood determined by the noise covariance. Moreover, if the $\mathcal{K}$-PL condition is strengthened to a $\mathcal{K}_\infty$-PL condition, the dynamics are NSS; whereas if it is weakened to a general positive-definite-PL condition, the dynamics exhibit integral NSS. The results further extend to objectives without globally Lipschitz gradients through appropriate step-size tuning. The proposed framework is further applied to the robustness analysis of policy optimization for the linear quadratic regulator (LQR) and logistic regression.

eess.SY↗

arXiv2025

Some remarks on gradient dominance and LQR policy optimization

Solutions of optimization problems, including policy optimization in reinforcement learning, typically rely upon some variant of gradient descent. There has been much recent work in the machine learning, control, and optimization communities applying the Polyak-Łojasiewicz Inequality (PLI) to such problems in order to establish an exponential rate of convergence (a.k.a. ``linear convergence'' in the local-iteration language of numerical analysis) of loss functions to their minima under the gradient flow. Often, as is the case of policy iteration for the continuous-time LQR problem, this rate vanishes for large initial conditions, resulting in a mixed globally linear / locally exponential behavior. This is in sharp contrast with the discrete-time LQR problem, where there is global exponential convergence. That gap between CT and DT behaviors motivates the search for various generalized PLI-like conditions, and this talk will address that topic. Moreover, these generalizations are key to understanding the transient and asymptotic effects of errors in the estimation of the gradient, errors which might arise from adversarial attacks, wrong evaluation by an oracle, early stopping of a simulation, inaccurate and very approximate digital twins, stochastic computations (algorithm ``reproducibility''), or learning by sampling from limited data. We describe an ``input to state stability'' (ISS) analysis of this issue. The second part discusses convergence and PLI-like properties of ``linear feedforward neural networks'' in feedback control. Much of the work described here was done in collaboration with Arthur Castello B. de Oliveira, Leilei Cui, Zhong-Ping Jiang, and Milad Siami.

cs.LG↗