arXiv subjects

Alexander Ovseevich

Publications and source records attributed to Alexander Ovseevich.

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.

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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.

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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.

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arXiv2023

On special quadratic Lyapunov functions for linear dynamical systems with an invariant cone

We consider a continuous-time linear time-invariant dynamical system that admits an invariant cone. For the case of a self-dual and homogeneous cone we show that if the system is asymptotically stable then it admits a quadratic Lyapunov function with a special structure. The complexity of this Lyapuonv function scales linearly with the dimension of the dynamical system. In the particular case when the cone is the nonnegative orthant this reduces to the well-known and important result that a positive system admits a diagonal Lyapunov function. We demonstrate our theoretical results by deriving a new special quadratic Lyapunov function for systems that admit the ice-cream cone as an invariant set.

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arXiv2023

Contraction and $k$-contraction in Lurie systems with applications to networked systems

A Lurie system is the interconnection of a linear time-invariant system and a nonlinear feedback function. We derive a new sufficient condition for $k$-contraction of a Lurie system. For $k=1$, our sufficient condition reduces to the standard stability condition based on the bounded real lemma and a small gain condition. However, Lurie systems often have more than a single equilibrium and are thus not contractive with respect to any norm. For $k=2$, our condition guarantees a well-ordered asymptotic behaviour of the closed-loop system: every bounded solution converges to an equilibrium, which is not necessarily unique. We demonstrate our results by deriving a sufficient condition for $k$-contraction of a general networked system, and then applying it to guarantee $k$-contraction in a Hopfield neural network, a nonlinear opinion dynamics model, and a 2-bus power system.

eess.SY↗

arXiv2023

A sufficient condition for $k$-contraction in Lurie systems

We consider a Lurie system obtained via a connection of a linear time-invariant system and a nonlinear feedback function. Such systems often have more than a single equilibrium and are thus not contractive with respect to any norm. We derive a new sufficient condition for $k$-contraction of a Lurie system. For $k=1$, our sufficient condition reduces to the standard stability condition based on the bounded real lemma and a small gain condition. For $k=2$, our condition guarantees well-ordered asymptotic behaviour of the closed-loop system: every bounded solution converges to an equilibrium, which is not necessarily unique. We apply our results to derive a sufficient condition for $k$-contractivity of a networked system.

eess.SY↗

arXiv2022

Verifying $k$-Contraction without Computing $k$-Compounds

Compound matrices have found applications in many fields of science including systems and control theory. In particular, a sufficient condition for $k$-contraction is that a logarithmic norm (also called matrix measure) of the $k$-additive compound of the Jacobian is uniformly negative. However, this may be difficult to check in practice because the $k$-additive compound of an $n\times n$ matrix has dimensions $\binom{n}{k}\times \binom{n}{k}$. For an $n\times n$ matrix $A$, we prove a duality relation between the $k$ and $(n-k)$ compounds of $A$. We use this duality relation to derive a sufficient condition for $k$-contraction that does not require the computation of any $k$-compounds. We demonstrate our results by deriving a sufficient condition for $k$-contraction of an $n$-dimensional Hopfield network that does not require to compute any compounds. In particular, for $k=2$ this sufficient condition implies that the network is $2$-contracting and this implies a strong asymptotic property: every bounded solution of the network converges to an equilibrium point, that may not be unique. This is relevant, for example, when using the Hopfield network as an associative memory that stores patterns as equilibrium points of the dynamics.

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arXiv2022

Asymptotic control theory for a closed string II

We develop an asymptotical control theory for one of the simplest distributed (infinite dimensional) oscillating systems, namely, for a closed string under a bounded load applied to a single distinguished point. We find exact classes of string states that admit complete damping and an asymptotically exact value of the required time. By using approximate reachable sets instead of exact ones, we design a feedback control, which turns out to be asymptotically optimal. The main results are an exact algebraic formula for asymptotic shape of the reachable sets, asymptotically optimal time of motion, and an asymptotically optimal control thus constructed.

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arXiv2019

Ribosome flow model with different site sizes

We introduce and analyze two general dynamical models for unidirectional movement of particles along a circular chain and an open chain of sites. The models include a soft version of the simple exclusion principle, that is, as the density in a site increases the effective entry rate into this site decreases. This allows to model and study the evolution of "traffic jams" of particles along the chain. A unique feature of these two new models is that each site along the chain can have a different size. Although the models are nonlinear, they are amenable to rigorous asymptotic analysis. In particular, we show that the dynamics always converges to a steady-state, and that the steady-state densities along the chain and the steady-state output flow rate from the chain can be derived from the spectral properties of a suitable matrix, thus eliminating the need to numerically simulate the dynamics until convergence. This spectral representation also allows for powerful sensitivity analysis, i.e. understanding how a change in one of the parameters in the models affects the steady-state. We show that the site sizes and the transition rates from site to site play different roles in the dynamics, and that for the purpose of maximizing the steady-state output (or production) rate the site sizes are more important than the transition rates. We also show that the problem of finding parameter values that maximize the production rate is tractable. We believe that the models introduced here can be applied to study various natural and artificial processes including ribosome flow during mRNA translation, the movement of molecular motors along filaments of the cytoskeleton, pedestrian and vehicular traffic, evacuation dynamics, and more.

q-bio.QM↗

arXiv2018

Networks of ribosome flow models for modeling and analyzing intracellular traffic

The ribosome flow model with input and output (RFMIO) is a deterministic dynamical system that has been used to study the flow of ribosomes during mRNA translation. The RFMIO and its variants encapsulate important properties that are relevant to modeling ribosome flow such as the possible evolution of "traffic jams" and non-homogeneous elongation rates along the mRNA molecule, and can also be used for studying additional intracellular processes such as transcription, transport, and more. Here we consider networks of interconnected RFMIOs as a fundamental tool for modeling, analyzing and re-engineering the complex mechanisms of protein production. In these networks, the output of each RFMIO may be divided, using connection weights, between several inputs of other RFMIOs. We show that under quite general feedback connections the network has two important properties: (1) it admits a unique steady-state and every trajectory converges to this steady-state, and (2) the problem of how to determine the connection weights so that the network steady-state output is maximized is a convex optimization problem. These mathematical properties make these networks highly suitable as models of various phenomena: property (1) means that the behavior is predictable and ordered, and property (2) means that determining the optimal weights is numerically tractable even for large-scale networks. For the specific case of a feed-forward network of RFMIOs we prove an additional useful property, namely, that there exists a spectral representation for the network steady-state, and thus it can be determined without any numerical simulations of the dynamics. We describe the implications of these result to several fundamental biological phenomena and biotechnological objectives.

q-bio.MN↗

arXiv2018

Asymptotic control theory for a closed string

We develop an asymptotical control theory for one of the simplest distributed oscillating systems, namely, for a closed string under a bounded load applied to a single distinguished point. We find exact classes of string states that admit complete damping and an asymptotically exact value of the required time. By using approximate reachable sets instead of exact ones, we design a dry-friction like feedback control, which turns out to be asymptotically optimal. We prove the existence of motion under the control using a rather explicit solution of a nonlinear wave equation. Remarkably, the solution is determined via purely algebraic operations. The main result is a proof of asymptotic optimality of the control thus constructed.

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arXiv2017

Optimal Translation Along a Circular mRNA

The ribosome flow model on a ring (RFMR) is a deterministic model for translation of a circularized mRNA. We derive a new spectral representation for the optimal steady-state production rate and the corresponding optimal steady-state ribosomal density in the RFMR. This representation has several important advantages. First, it provides a simple and numerically stable algorithm for determining the optimal values even in very long rings. Second, it enables efficient computation of the sensitivity of the optimal production rate to small changes in the transition rates along the mRNA. Third, it implies that the optimal steady-state production rate is a strictly concave function of the transition rates. Thus maximizing the optimal steady-state production rate with respect to the rates, under an affine constraint on the rates becomes a convex optimization problem that admits a unique solution, which can be determined numerically using highly efficient algorithms. This optimization problem is important, for example, when re-engineering heterologous genes in a host organism. We describe the implications of our results to this and other aspects of translation.

q-bio.SC↗

arXiv2017

Asymptotically optimal control for a simplest distributed system

We study the problem of the minimum-time damping of a closed string under a bounded load, applied at a single fixed point. A constructive feedback control law is designed, which allows bringing the system to a bounded neighbourhood of the terminal manifold. The law has the form of the dry friction at the point, where the load is applied. The motion under the control is governed by a nonlinear wave equation. The existence and uniqueness of solution of the Cauchy problem for this equation are proved. The main result is the asymptotic optimality of the suggested control law.

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arXiv2016

Asymptotically optimal feedback control for a system of linear oscillators

Problem of damping of an arbitrary number of linear oscillators under common bounded control is considered. We are looking for a feedback control steering the system to the equilibrium. The obtained control is asymptotically optimal: the ratio of motion time to zero with this control to the minimum one is close to 1, if the initial energy of the system is large. Some of the results based on the new lemma about sustainable observability autonomous linear systems.

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arXiv2016

Damping of a system of linear oscillators using the generalized dry friction

The problem of damping a system of linear oscillators is considered. The problem is solved by using a control in the form of dry friction. The motion of the system under the control is governed by a system of differential equations with discontinuous right-hand side. A uniqueness and continuity theorem is proved for the phase flow of this system. Thus, the control in the form of generalized dry friction defines the motion of the system of oscillators uniquely.

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arXiv2016

Asymptotic control theory for a system of linear oscillators

We present an asymptotic control theory for a system of an arbitrary number of linear oscillators under a common bounded control. We suggest a design method of a feedback control for this system. By using the DiPerna-Lions theory of singular ODEs, we prove that the suggested control law correctly defines the motion of the system. The obtained control is asymptotically optimal: the ratio of the motion time to zero under this control to the minimum one is close to 1 if the initial energy of the system is large. The results are partially based on a new perturbation theory of observable linear systems.

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arXiv2015

Feedback control for damping a system of linear oscillators

The paper is devoted to a design of a common bounded feedback control steering a system of an arbitrary number of linear oscillators to the equilibrium. At high energies, the control is based on the asymptotic theory of reachable sets of linear systems. With decreasing of the energy, a similar control with a reduced upper bound is used. On the final stage, the control is constructed by using the method of common Lyapunov functions. Special attention is paid to the cases of one and two oscillators.

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arXiv2015

Perturbation theory of observable linear systems

The present work is motivated by the asymptotic control theory for a system of linear oscillators: the problem is to design a common bounded scalar control for damping all oscillators in asymptotically minimal time. The motion of the system is described in terms of a canonical system similar to that of the Pontryagin maximum principle. We consider the evolution equation for adjoint variables as a perturbed observable linear system. Due to the perturbation, the unobservable part of the state trajectory cannot be recovered exactly. We estimate the recovering error via the $L_1$-norm of perturbation. This allows us to prove that the control makes the system approach the equilibrium state with a strictly positive speed.

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