arXiv subjects

Alexander Zuyev

Publications and source records attributed to Alexander Zuyev.

At least 19 recordsLinked to original sources

arXiv2026

Boundary control and periodic trajectories of semilinear Euler equations with application to hydrogen transport

A mathematical model of gas flow in a pipeline controlled by the inlet pressure and the outlet mass flow flux is considered in the form of the isothermal Euler equations with an appropriate equation of state. Within the framework of boundary control systems, this model is transformed into a nonlinear abstract differential equation using an appropriate lifting operator. For this abstract equation, a representation in terms of Fourier coefficients is derived analytically. Conditions for the existence of periodic solutions to a broad class of nonlinear control systems with continuously differentiable controls are established in abstract spaces. This framework is applied to a realistic hydrogen transport model to evaluate periodic operating regimes under periodic fluctuations in supply and demand.

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arXiv2026

Analytical and Reduced-Order Modeling of a Timoshenko Beam with Point Actuation

We present frequency-domain models of a controlled mechanical system consisting of a flexible beam and a rigid body. The transfer function for the beam, governed by the Timoshenko equations under the body-beam interface conditions, is derived analytically. Based on this infinite-dimensional representation, reduced-order models are constructed within the framework of Loewner matrices. A comparative analysis of the Bode plots is presented for the beam model using different choices of sensor-actuator pairs to validate the proposed frequency-domain modeling framework.

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arXiv2026

Exponential Decay for a Boundary-Controlled Nonlinear Parabolic Reactor Model

We study an axial dispersion tubular reactor model governed by a nonlinear parabolic equation with Robin-type boundary conditions and boundary feedback control. We derive sufficient conditions for the exponential stability of the steady-state solution of the closed-loop system and provide an explicit estimate of the decay rate. In addition, numerical simulations are presented to illustrate the sharpness of the obtained decay rate for different choices of the feedback gain parameter.

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arXiv2026

Periodic solutions of nonlinear control systems with switching: a Lie-algebraic and contraction approach

This paper is devoted to the analysis of periodic solutions of nonlinear control-affine systems with bang-bang controls. Such problems naturally arise in periodic optimal control with constrained inputs, which have, in particular, important applications in the performance optimization of chemical reactions. We reduce the problem of constructing a periodic solution to that of finding a fixed point of a composition of exponential maps. The latter problem is then addressed using the Baker-Campbell-Hausdorff-Dynkin (BCHD) formula. We establish the equivalence between periodic solutions of the original control system and those of an associated autonomous system involving iterated Lie brackets. Applying incremental stability arguments allows us to further simplify the problem to finding the equilibria of this autonomous system. The developed theory is then applied to nonlinear chemical reaction models with constrained controls.

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arXiv2026

Data-Driven Modeling of a Controlled Orthotropic Plate Using Machine Learning

We study the problem of learning the input-output map of a controlled vibrating plate with a composite structure from experimental measurements. Analytical modeling of this control system faces challenges due to the essential orthotropy and unknown damping characteristics of the material. Surrogate models based on linear regression, multilayer perceptrons, and gated recurrent units are constructed from the available sampled data. Through comparative analysis, we show that the multilayer perceptron model provides an acceptable approximation of this dynamical system, capturing the potentially nonlinear phenomena in its input-output behavior.

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arXiv2026

Polynomial Turnpike Property for a Class of Infinite-Dimensional Oscillating Systems

We establish a polynomial turnpike estimate for an optimal control problem consisting of a system of infinitely many controlled oscillators, considered as an abstract differential equation in a Hilbert space, with a quadratic cost. Our proof relies on spectral considerations and on the construction of a Riesz basis. A concrete example is given, which involves a rotating bodybeam system. To our knowledge, this is the first example of a pointwise turnpike estimate around a steady-state that is polynomial but not exponential.

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arXiv2026

Implementation of Time-Varying Controllers for a Nonholonomic Mobile Robot: Experimental Studies

We consider a kinematic model of a wheeled mobile robot controlled by translational and angular velocities. For this class of nonholonomic systems, a family of time-varying feedback controllers was proposed in our previous works using gradient flow approximation techniques. In the present study, these controllers are implemented on a TurtleBot3 Burger (TB3) mobile robot to provide experimental validation of the stabilization problem with oscillating input signals. In addition, the admissibility problem of a gradient flow is investigated to justify the construction of a Lyapunov function candidate. The presented experimental results demonstrate the possibility of stabilizing the reference position of the robot using feedback controls with practically acceptable parameters.

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arXiv2025

Frequency-Domain Analysis of the Euler-Bernoulli and Timoshenko Beams with Attached Masses

This work focuses on the frequency-domain modeling of a control system with a flexible beam and a rigid body. A simply supported beam is equipped with a spring-loaded control actuator and possesses local damping effect. Using Hamilton's variational principle, the equations of motion are derived in the state space form taking into account interface conditions involving lumped control and local damping. The transfer functions are obtained for the Timoshenko and Euler--Bernoulli beam models with the output measurements provided by a point sensor. Comparative Bode plots are presented for the two beam models with different choices of output signals and damping coefficients.

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arXiv2025

A Model-Free Extremum Seeking Controller with Application to Tracking a Nonlinear Chemical Reaction

In this paper, we develop the extremum-seeking approach to generate admissible trajectories in a neighborhood of a given reference curve in the state space. The cost function of the problem represents the distance between the current system state and the reference curve, which is parameterized as a function of time. Such reference curves naturally arise as optimal trajectories in isoperimetric optimization problems for nonlinear chemical reactions, where the objective is to maximize the average reaction product over a given period. We apply the proposed extremum seeking control design to a nonisothermal reaction model and illustrate the resulting tracking errors through numerical simulations.

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arXiv2025

On classical solutions in the stabilization problem for nonholonomic control systems with time-varying feedback laws

We consider the stabilization problem for driftless control-affine systems under the bracket-generating condition. In our previous works, a class of time-varying feedback laws has been constructed to stabilize the equilibrium of a nonholonomic system under rather general controllability assumptions. This stabilization scheme is based on the sampling concept, which is not equivalent to the classical definition of solutions for the corresponding nonautonomous closed-loop system. In the present paper, we refine the previous results by presenting sufficient conditions for the convergence of classical solutions of the closed-loop system to the equilibrium. Our theoretical findings are applied to a multidimensional driftless control-affine system and illustrated through numerical simulations.

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arXiv2024

Stability and decay rate estimates for a nonlinear dispersed flow reactor model with boundary control

We investigate a nonlinear parabolic partial differential equation whose boundary conditions contain a single control input. This model describes a chemical reaction of the type ``$A \to $ product'', occurring in a dispersed flow tubular reactor. The existence and uniqueness of solutions to the nonlinear Cauchy problem under consideration are established by applying the theory of strongly continuous semigroups of operators. We also prove the stability of the equilibrium of the closed-loop system with a proposed feedback law. Additionally, using Lyapunov's direct method, we evaluate the exponential decay rate of the solutions.

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arXiv2024

Polynomial Convergence of an Observer for an Infinite-Dimensional Oscillating System

This paper is devoted to analyzing the observer convergence rate for a class of linear control systems in a Hilbert space. To characterize the polynomial stability of the observer error system, we apply the spectral theory of linear operators and explicitly construct the resolvent of the corresponding infinitesimal generator. The asymptotic behavior of the resolvent on the imaginary axis is studied to describe the rate of decay of the observation error. The estimated decay rate is illustrated through an example of an oscillating flexible structure with one-dimensional output.

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arXiv2024

On the Controllability of an Orbiting Satellite Model with Electromagnetic-only Actuation

This paper presents sufficient conditions for small-time local controllability of a control-affine system that describes the rotational motion of a satellite in a circular orbit. The satellite is modeled as a rigid body subject to electromagnetic actuation. We focus on the underactuated scenario where the control torque is generated solely by magnetorquers. The main contributions of this work include proving small-time local controllability around the relative equilibrium under some natural assumptions on the mass distribution of the rigid body. This result is based on the Lie algebra rank condition and Sussmann's controllability condition. Furthermore, it is shown that the linearized system is not controllable in a neighborhood of the considered equilibrium.

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arXiv2024

Dynamic Modelling of a Controlled Orthotropic Plate: Analytic and Data-Driven Approaches in the Frequency Domain

This paper is devoted to the mathematical modelling of a vibrating orthotropic plate equipped with a laminated piezosensor, under the influence of a lumped force actuation. We employ the Kirchhoff plate theory to derive the corresponding partial differential equation, assuming free boundary conditions. Analytical solutions for this boundary value problem are explored in the form of series expansions, using products of Krylov functions. Utilizing Galerkin's method, this mathematical model is transformed into an infinite-dimensional control system characterized by modal coordinates. The transfer function of such a system is explicitly evaluated in the single-input single-output case. The computation of coefficients for finite-dimensional approximate systems is formalized in an algorithm with an arbitrary number of degrees of freedom. Our numerical study confirms that the modeled input-output behavior shows acceptable agreement over the given frequency range.

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arXiv2024

Estimates of the Kolmogorov n-width for nonlinear transformations with application to distributed-parameter control systems

This paper aims at characterizing the approximability of bounded sets in the range of nonlinear operators in Banach spaces by finite-dimensional linear varieties. In particular, the class of operators we consider includes the endpoint maps of nonlinear distributed-parameter control systems. We describe the relationship between the Kolmogorov n-width of a bounded subset and the width of its image under an essentially nonlinear transformation. We propose explicit estimates of the n-width in the space of images in terms of the affine part of the corresponding operator and the width of its nonlinear perturbation. These $n$-width estimates enable us to describe the reachable sets for infinite-dimensional bilinear control systems, with applications to controlling the Euler-Bernoulli beam using a contraction force and to a single-input Schr\"odinger equation.

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arXiv2024

Stabilization of a Nonholonomic Car Model with Off-Hooked Trailers

We consider a kinematic model of a controlled car with two trailers by assuming that each trailer is attached at some distance from the preceding axle ("off-hooked trailers"). For this model, we derive the transformation towards privileged coordinates and present the corresponding nilpotent quasihomogeneous approximate system. The components of this nilpotent approximation are written explicitly in terms of mechanical parameters of the original system. The constructed system does not satisfy the Brockett necessary stabilizability condition, and the design of time-varying feedback controllers with oscillating components is proposed. It is proved that these controllers ensure the exponential convergence of solutions to the trivial equilibrium, and simulation results are presented to illustrate the behavior of the closed-loop system.

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arXiv2023

Iterative approximations of periodic trajectories for nonlinear systems with discontinuous inputs

Nonlinear control-affine systems described by ordinary differential equations with bounded measurable input functions are considered. The solvability of general boundary value problems for these systems is formulated in the sense of Carath\'eodory solutions. It is shown that, under the dominant linearization assumption, the considered class of boundary value problems admits a unique solution for any admissible control. These solutions can be obtained as the limit of the proposed simple iterative scheme and, in the case of periodic boundary conditions, via the developed Newton-type schemes. Under additional technical assumptions, sufficient contraction conditions of the corresponding generating operators are derived analytically. The proposed iterative approach is applied to compute periodic solutions of a realistic chemical reaction model with discontinuous control inputs.

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arXiv2023

Partial Stabilization of an Orbiting Satellite Model with a Flexible Attachment

We consider a mathematical model of an orbiting satellite, comprising a rigid carrier body and a flexible boom, operating under the influence of gravity gradient torque. This model is represented by a nonlinear control system, which includes ordinary differential equations governing the carrier body's angular velocity and attitude quaternion, coupled with the Euler-Bernoulli equations that describe the vibration of the flexible component. We propose an explicit feedback design aimed at guaranteeing the partial stability of the closed-loop system in an appropriate Hilbert space.

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