arXiv subjects

Michael Margaliot

Publications and source records attributed to Michael Margaliot.

At least 19 recordsLinked to original sources

arXiv2026

On the Asymptotic Switching Density in Time-Optimal Control of Linear Systems

We study the time-optimal control of a controllable linear system on a time horizon [0,T], focusing on the asymptotic switching density for large T. When the system matrix has only real eigenvalues, it is well-known that the number of switches is upper bounded uniformly in T; when it has complex eigenvalues, no such uniform bound exists, and the switching count instead typically grows with T. We characterize this growth for a system matrix with an arbitrary spectrum, allowing simultaneously for real eigenvalues, complex eigenvalues, and a non-trivial Jordan structure. If the dominant mode is complex, the number of switches grows at least linearly in T, with an explicit lower bound expressed via the mean motion problem and the Bohl-Weyl-Wintner formula. We illustrate the theory on a linearized aircraft pitch/altitude model, showing close agreement between the predicted asymptotic switching rate and numerically computed time-optimal controls.

math.OC↗

arXiv2026

Gain of Entrainment in Nonlinear Cascades

We consider the gain of entrainment (GOE)--the difference between the average steady-state output under a periodic input and the steady-state output under a constant input with the same mean--for an $n$-stage feedforward cascade of stable first-order filters interleaved with static nonlinearities. The main result is an exact decomposition of GOE as a weighted sum of local Jensen gaps, where each gap quantifies the mean shift generated by a nonlinearity, and each weight is a product of downstream incremental gains divided by linear time constants. We provide a Bregman-divergence interpretation of the decomposition, and a second-order small-amplitude of GOE separating local curvature, fluctuation energy, and differential gains. We demonstrate the theoretical results using a Michaelis-Menten cascade showing that any nonconstant periodic feeding strictly reduces the average terminal product relative to constant feeding with the same mean.

math.OC↗

arXiv2026

On the Cost of Entrainment in Protein Translation

Biological systems often synchronize their dynamics with periodic environmental and intracellular signals. Whether such periodic coordination can also improve performance, however, remains unclear. Here, we study this question in the ribosome flow model, a nonlinear dynamical model of ribosome movement along an mRNA transcript during translation. We compare the average protein production rate under positive periodic transition rates with that of a constant-rate system obtained by replacing each rate by its temporal average. We prove that periodic modulation can never increase average protein production: the gain of entrainment is always nonpositive. Moreover, the gain is zero if and only if all transition rates share a common positive periodic modulation of their mean values. This exceptional modulation merely reparametrizes time and leaves the trajectory through state space unchanged. Thus, any genuinely nonuniform temporal modulation strictly reduces average protein production. Our results establish a fundamental tradeoff between temporal coordination and translational efficiency: entrainment can synchronize translation with periodic cellular programs, but this synchronization comes at a quantifiable production cost.

q-bio.MN↗

arXiv2026

A universal multi-turnpike principle for optimal allocation of translational resources

mRNA translation in the cell requires efficient allocation of shared and limited resources including free ribosomes, tRNA molecules, and initiation factors across multiple transcripts. Using a network of dynamic mathematical models for ribosome flow along the mRNA, we pose the problem of maximizing the total steady-state protein production rate in the cell under a shared and limited total budget for all translation rates in all the transcripts. We prove that the optimal solution of this resource allocation problem admits a multi-turnpike structure: in each mRNA, the transition rates are high and nearly uniform along the bulk of the coding region, with lower and varying rates near the boundaries of the~mRNA. Our results are based on the emergence of hierarchical optimality: regardless of how resources are allocated among genes, every transcript should internally organize itself in essentially the same way. This suggests that to optimize the overall production rate it is sufficient to regulate the initiation and termination regions in each transcript. Remarkably, this universal turnpike structure holds for any number of transcripts, arbitrary transcript lengths, and various optimization criteria.This agrees with observed conserved translational phenomena, such as codon ramps and initiation-dominated regulation. Our findings may also provide guidelines for the rational design of intracellular circuits operating under translational control.

q-bio.MN↗

arXiv2026

On the gain of entrainment in stable linear control systems with a nonlinear output

A control system admits a positive gain of entrainment (GOE) if entrainment to a periodic input yields a larger output, on average, than the output generated by the corresponding constant input with the same mean value. We analyze GOE in continuous-time stable linear control systems with a static nonlinear output map. Although linear systems with linear outputs have zero GOE, we show that a nonlinear output may generate a nontrivial GOE through the mismatch between the average output along the entrained periodic orbit and the output evaluated at the corresponding averaged equilibrium. We derive a second-order characterization of GOE for smooth output maps revealing that the leading-order contribution is determined by the curvature of the output map. We then show that if the output is convex (concave) on the controllable subspace, then GOE is nonnegative (nonpositive) for every periodic input. Furthermore, GOE admits a natural geometric interpretation as the average Bregman divergence between the entrained periodic orbit and the equilibrium associated with the averaged input. For the special case of quadratic output functions, we derive explicit frequency-domain formulas for GOE. These yield necessary and sufficient conditions guaranteeing the sign of GOE, characterize the contribution of individual input harmonics, and lead to an optimal periodic excitation that maximizes GOE under an energy constraint. The theoretical results are illustrated using an electrical RLC circuit and a compartmental pharmacodynamic model with a nonlinear drug-effect map.

math.OC↗

arXiv2026

Orthant-Monotonic Norms and Additive D-Stability

Matrix measures induced by vector norms are widely used in contraction theory of nonlinear dynamical systems. A natural and important robustness question is whether negativity of a matrix measure is preserved under arbitrary nonnegative diagonal damping. Matrix measures with this property have been called admissible. We show that an induced matrix measure is admissible if and only if the underlying norm is orthant-monotonic. Equivalently, these are precisely the induced matrix measures satisfying $\mu(D) = \max_i\{d_{ii} \}$ for every nonnegative diagonal matrix $D$. We further show that this class is maximal for uniform preservation of contraction under nonnegative diagonal perturbations. The result gives a new geometric characterization of admissibility and clarifies the role of orthant-monotonicity in additive $D$-stability and diffusion-induced instability.

math.OC↗

arXiv2026

Structure preserving properties of higher order moment closures for TASEP

The totally asymmetric simple exclusion process (TASEP) is a stochastic model for the unidirectional flow of interacting particles on a 1D-lattice that is much used in systems biology and statistical physics. Its master equation describes the evolution of the probability distribution on the configuration space. The size of the master equation grows exponentially with the length of the lattice. It is known that the complexity of the system may be reduced using mean-field approximations. We provide a rigorous definition of a family of such models using moments of any order and an extension to the pair approximation for obtaining closures for the system. The dimension of these models grows linearly with the lattice size and exponentially in the order of the approximation. Moreover, we show that the states of these models still have a probabilistic interpretation and that basic structural properties of the master equation are preserved. This extends known results on the Ribosome Flow Model which can be viewed as the first order approximation for TASEP.

math.DS↗

arXiv2026

A turnpike property in an eigenvalue optimization problem

We consider a constrained eigenvalue optimization problem that arises in an important nonlinear dynamical model for mRNA translation in the cell. We prove that the ordered list of optimal parameters admits a turnpike property, namely, it includes three parts with the first and third part relatively short, and the values in the middle part are all approximately equal. Turnpike properties have attracted considerable attention in econometrics and optimal control theory, but to the best of our knowledge this is the first rigorous proof of such a structure in an eigenvalue optimization problem.

math.OC↗

arXiv2025

Guardian maps for continuous-time systems: A Lie-algebraic approach

Guardian maps are scalar maps that vanish when a matrix or polynomial is on the verge of stability. Several guardian maps have been proposed in the literature for Hurwitz stability based on the Kronecker sum, the second lower Schl\"aflian matrix, and the bialternate sum. It is natural to ask if there is a unifying principle for all these maps. Here, we introduce the Lie-algebraic notion of a guardian representation, and show that all the examples above are instances of this unifying idea. We also show that the bialternate sum coincides with the second additive compound.

math.OC↗

arXiv2025

Negative feedback and oscillations in a model for mRNA translation

The ribosome flow model (RFM) is a phenomenological model for the unidirectional flow of particles along a 1D chain of $n$ sites. The RFM has been extensively used to study the dynamics of ribosome flow along a single mRNA molecule during translation. In this case, the particles model ribosomes and each site corresponds to a consecutive group of codons. Networks of interconnected RFMs have been used to model and analyze large-scale translation in the cell and, in particular, the effects of competition for shared resources. Here, we analyze the RFM with a negative feedback connection from the protein production rate to the initiation rate. This models, for example, the production of proteins that inhibit the translation of their own mRNA. Using tools from the theory of 2-cooperative dynamical systems, we provide a simple condition guaranteeing that the closed-loop system admits at least one non-trivial periodic solution. When this condition holds, we also explicitly characterize a large set of initial conditions such that any solution emanating from this set converges to a non-trivial periodic solution. Such a solution corresponds to a periodic pattern of ribosome densities along the mRNA, and to a periodic pattern of protein production.

q-bio.MN↗

arXiv2025

Instability of equilibrium and convergence to periodic orbits in strongly 2-cooperative systems

We consider time-invariant nonlinear $n$-dimensional strongly $2$-cooperative systems, that is, systems that map the set of vectors with up to weak sign variation to its interior. Strongly $2$-cooperative systems enjoy a strong Poincare-Bendixson property: bounded solutions that maintain a positive distance from the set of equilibria converge to a periodic solution. For strongly $2$-cooperative systems whose trajectories evolve in a bounded and invariant set that contains a single unstable equilibrium, we provide a simple criterion for the existence of periodic trajectories. Moreover, we explicitly characterize a positive-measure set of initial conditions which yield solutions that asymptotically converge to a periodic trajectory. We demonstrate our theoretical results using two models from systems biology, the $n$-dimensional Goodwin oscillator and a $4$-dimensional biomolecular oscillator with RNA-mediated regulation, and provide numerical simulations that verify the theoretical results.

math.DS↗

arXiv2025

An application of the mean motion problem to time-optimal control

We consider time-optimal controls of a controllable linear system with a scalar control on a long time interval. It is well-known that if all the eigenvalues of the matrix describing the linear system dynamics are real then any time-optimal control has a bounded number of switching points, where the bound does not depend on the length of the time interval. We consider the case where the governing matrix has purely imaginary eigenvalues, and show that then, in the generic case, the number of switching points is bounded from below by a linear function of the length of the time interval. The proof is based on relating the switching function in the optimal control problem to the mean motion problem that dates back to Lagrange and was solved by Hermann Weyl.

math.OC↗

arXiv2025

Random attraction in TASEP with time-varying hopping rates

The totally asymmetric simple exclusion principle (TASEP) is a fundamental model in nonequilibrium statistical mechanics. It describes the stochastic unidirectional movement of particles along a 1D chain of ordered sites. We consider the continuous-time version of TASEP with a finite number of sites and with time-varying hopping rates between the sites. We show how to formulate this model as a nonautonomous random dynamical system (NRDS) with a finite state-space. We provide conditions guaranteeing that random pullback and forward attractors of such an NRDS exist and consist of singletons. In the context of the nonautonomous TASEP these conditions imply almost sure synchronization of the individual random paths. This implies in particular that perturbations that change the state of the particles along the chain are "filtered out" in the long run. We demonstrate that the required conditions are tight by providing examples where these conditions do not hold and consequently the forward attractor does not exist or the pullback attractor is not a singleton. The results in this paper generalize our earlier results for autonomous TASEP in https://doi.org/10.1137/20M131446X and contain these as a special case.

math.OC↗

arXiv2024

A networked small-gain theorem based on discrete-time diagonal stability

We present a new sufficient condition for finite-gain $L_2$ input-to-output stability of a networked system. The condition requires a matrix, that combines information on the $L_2$ gains of the sub-systems and their interconnections, to be discrete-time diagonally stable (DTDS). We show that the new result generalizes the standard small gain theorem for the negative feedback connection of two sub-systems. An important advantage of the new result is that known sufficient conditions for DTDS can be applied to derive sufficient conditions for networked input-to-output stability. We demonstrate this using several examples. We also derive a new necessary and sufficient condition for a matrix that is a rank one perturbation of a Schur diagonal matrix to be DTDS.

eess.SY↗

arXiv2024

A sufficient condition for 2-contraction of a feedback interconnection

Multistationarity - the existence of multiple equilibrium points - is a common phenomenon in dynamical systems from a variety of fields, including neuroscience, opinion dynamics, systems biology, and power systems. A recently proposed generalization of contraction theory, called $k$-contraction, is a promising approach for analyzing the asymptotic behaviour of multistationary systems. In particular, all bounded trajectories of a time-invariant 2-contracting system converge to an equilibrium point, but the system may have multiple equilibrium points where more than one is locally stable. An important challenge is to study $k$-contraction in large-scale interconnected systems. Inspired by a recent small-gain theorem for 2-contraction by Angeli et al., we derive a new sufficient condition for 2-contraction of a feedback interconnection of two nonlinear dynamical systems. Our condition is based on (i) deriving new formulas for the 2-multiplicative [2-additive] compound of block matrices using block Kronecker products [sums], (ii) a hierarchical approach for proving standard contraction, and (iii) a network small-gain theorem for Metzler matrices. We demonstrate our results by deriving a simple sufficient condition for 2-contraction in a network of FitzHugh-Nagumo neurons.

eess.SY↗

arXiv2024

Model order reduction for the TASEP Master equation

The totally asymmetric simple exclusion process (TASEP) is a stochastic model for the unidirectional dynamics of interacting particles on a $1$D-lattice that is much used in systems biology and statistical physics. Its master equation describes the evolution of the probability distribution on the state space. The size of the master equation grows exponentially with the length of the lattice. It is known that the complexity of the system may be reduced using mean field approximations. We provide a rigorous derivation and a stochastic interpretation of these approximations and present numerical results on their accuracy for a number of relevant cases.

cond-mat.stat-mech↗

arXiv2024

Convergence to periodic orbits in 3-dimensional strongly 2-cooperative systems

The flow of a $k$-cooperative system maps the set of vectors with up to~$(k-1)$ sign variations to itself. Strongly $2$-cooperative systems satisfy a strong \Poincare-Bendixson property: any bounded solution that evolves in a compact set containing no equilibria converges to a periodic orbit. For $3$-dimensional strongly $2$-cooperative nonlinear systems, we provide a simple sufficient condition that guarantees the existence, in the state space, of an invariant compact set that includes no equilibrium points. Thus, any solution emanating from this set converges to a periodic orbit. We characterize explicitly the set of initial conditions from which the trajectory converges to a periodic solution. We demonstrate our theoretical results on two well-known models in biochemistry: a 3D Goodwin oscillator model and the 3D Field-Noyes ordinary-differential-equation (ODE) model for the Belousov-Zhabotinskii reaction.

math.OC↗

arXiv2024

Analysis of the Identifying Regulation with Adversarial Surrogates Algorithm

Given a time-series of noisy measured outputs of a dynamical system z[k], k=1...N, the Identifying Regulation with Adversarial Surrogates (IRAS) algorithm aims to find a non-trivial first integral of the system, namely, a scalar function g() such that g(z[i]) = g(z[j]), for all i,j. IRAS has been suggested recently and was used successfully in several learning tasks in models from biology and physics. Here, we give the first rigorous analysis of this algorithm in a specific setting. We assume that the observations admit a linear first integral and that they are contaminated by Gaussian noise. We show that in this case the IRAS iterations are closely related to the self-consistent-field (SCF) iterations for solving a generalized Rayleigh quotient minimization problem. Using this approach, we derive several sufficient conditions guaranteeing local convergence of IRAS to the correct first integral.

eess.SY↗