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## О задаче граничного управления для системы, описываемой двумерным волновым уравнением

Известия РАН. Теория и системы управления. 2019. № 1. С. 109-116.

Romanov I., Шамаев А. С.

The boundary controllability of oscillations of a plane membrane is studied. The magnitude

of the control is bounded. The controllability problem of driving the membrane to rest is considered.

The method of proof proposed in this paper can be applied to any dimension but only the two-dimen-

sional case is considered for simplicity.

Romanov I., Shamaev A., Journal of Optimization Theory and Applications 2021 Vol. 188 No. 3 P. 925-938

The problem of the exact bounded control of oscillations of the two-dimensional membrane is considered. Control force is applied to the boundary of the membrane, which is located in a domain on a plane. The goal of the
control is to drive the system to rest in finite time. ...

Added: May 23, 2021

Vexler B., Zlotnik A., Trautmann P., Comptes Rendus Mathematique 2018 Vol. 356 No. 5 P. 523-531

The paper deals with the optimal control problems governed by the 1D wave equation with variable coefficients and the control spaces of either measure-valued functions or vector measures. Bilinear finite element discretizations are constructed and their stability and error analysis is accomplished. ...

Added: April 8, 2017

Zlotnik A., Kireeva O., / Cornell University. Series arXiv "math". 2020. No. arXiv:2011.14104v2[math.NA].

We consider compact finite-difference schemes of the 4th approximation order for an initial-boundary value problem (IBVP) for the $n$-dimensional non-homogeneous wave equation, $n\geq 1$. Their construction is accomplished by both the classical Numerov approach and alternative technique based on averaging of the equation, together with further necessary improvements of the arising scheme for $n\geq 2$. The ...

Added: December 1, 2020

Romanov I., Shamaev A., Journal of Computer and Systems Sciences International 2020 Vol. 59 No. 3 P. 371-380

The problem of the exact bounded control of transverse vibrations of a thin plate is considered. Control actions are applied to the boundary of the plate, which fills a certain bounded domain on the plane. The purpose of the control is to completely stop oscillations in a finite time period. ...

Added: August 9, 2020

Trautmann P., Vexler B., Zlotnik A., Mathematical Control and Related Fields 2018 Vol. 8 No. 2 P. 411-449

This work is concerned with the optimal control problems governed by the 1D wave equation with variable coefficients and the control spaces $\mathcal M_T$ of either measure-valued functions $L^2(I,\mathcal M(\Omega))$ or vector measures $\mathcal M(\Omega,L^2(I))$. The cost functional involves the standard quadratic terms and the regularization term $\alpha\|u\|_{\mathcal M_T}$, $\alpha>0$. We construct and study three-level ...

Added: April 8, 2017

Maslov V., Теоретическая и математическая физика 2021 Т. 206 № 3 С. 448-452

We consider the construction of asymptotic solutions of linear equations related to equations of classical mechanics: the Hamilton–Jacobi equation and the transport equation. We show that these methods and also the theory of the mechanics of an infinitely narrow beam as a whole can be applied to some objects in bioenergy if the thin organic ...

Added: July 14, 2021

Romanov I., Shamaev A., / Cornell University. Series math "arxiv.org". 2018. No. arXiv:1603.01212v3.

The problem of the exact bounded control of oscillations of the two-dimensional membrane is considered. Control force is applied to the boundary of the membrane, which is located in a domain on a plane. The goal of the control is to drive the system to rest in a finite time. ...

Added: October 19, 2016

Zlotnik A., Čiegis R., Applied Mathematics and Computation 2022 Vol. 412 Article 126565

We consider an initial-boundary value problem for the $n$-dimensional wave equation, $n\geq 2$, with the variable sound speed with the nonhomogeneous Dirichlet boundary conditions. We construct and study three-level in time and compact in space three-point in each spatial direction alternating direction implicit (ADI) schemes having the approximation orders $\mathcal{O}(h_t^2+|h|^4)$ and $\mathcal{O}(h_t^4+|h|^4)$ on the uniform rectangular mesh.
The study includes stability ...

Added: August 11, 2021

Бекларян Л. А., Beklaryan A., Belousov F., Вестник Тамбовского университета. Серия: Естественные и технические науки 2015 Т. 20 № 6 С. 1736-1747

We investigate a problem of the existence of traveling-wave-type solutions for the finite-difference analogue of a nonlinear wave equation. In case of an inhomogeneous medium for vanishing traveling-wave-type solutions a natural extension in the form of quasi-traveling-wave-type solutions is given. ...

Added: December 7, 2015

Romanov I., Shamaev A., / Cornell University. Series math "arxiv.org". 2016. No. arXiv:1408.0382.

We will consider the exact controllability of the distributed system governed by two-dimensional Gurtin-Pipkin equation. It will be proved that this mechanical system can not be driven to an equilibrium point in a finite time if the distributed control has a compact support properly contained in the considered domain. In this case the memory kernel ...

Added: October 4, 2014

Zlotnik A., Čiegis R., / Cornell University. Series arXiv "math". 2021. No. ArXiv: 2101.10575v2[math.NA].

We consider an initial-boundary value problem for the $n$-dimensional wave equation with the variable sound speed, $n\geq 1$. We construct three-level implicit in time compact in space (three-point in each space direction) 4th order finite-difference schemes on the uniform rectangular meshes including their one-parameter (for $n=2$) and three-parameter (for $n=3$) families. They are closely connected to some ...

Added: February 2, 2021

Beklaryan L. A., Beklaryan A., Computational Mathematics and Mathematical Physics 2021 Vol. 61 No. 12 P. 1980-1994

The existence of a family of bounded soliton solutions for a finite-difference wave equation with a quadratic potential is established. The proof is based on a formalism establishing a one-to-one correspondence between the soliton solutions of an infinite-dimensional dynamical system and the solutions of a family of functional differential equations of the pointwise type. A ...

Added: January 14, 2022

Zlotnik A., / Cornell University. Series arXiv "math". 2021. No. 2105.07206.

Added: August 7, 2021

Бекларян Л. А., Beklaryan A., Журнал вычислительной математики и математической физики 2021 Т. 61 № 12 С. 2024-2039

The existence of a family of bounded soliton solutions for a finite difference wave equation with a quadratic potential is established. The proof is carried out within the framework of a formalism that establishes a one-to-one correspondence between soliton solutions of an infinite-dimensional dynamical system and solutions of a family of pointwise functional differential equations. For ...

Added: November 1, 2021

Romanov I., Romanova A., IFAC-PapersOnLine 2018 Vol. 51 No. 2 P. 132-137

The article is devoted to the problems of controllability of some systems with memory. These systems are described by partial integrodifferential equations. Null controllability and controllability to rest of the solutions are considered. In contrast to partial differential equations, these terms are not identical to each other. ...

Added: June 30, 2018

Manita L., Современная математика и ее приложения 2011 Т. 69

Рассматриваются задачи минимизации квадратичного функционала на траекториях волнового уравнения, которые на основе метода Фурье сводятся к задаче управления счетной системой обыкновенных дифференциальных уравнений. В качестве управления выступает плотность внешних сил. Исследуется проблема управляемости. ...

Added: October 7, 2013

Trautmann P., Vexler B., Zlotnik A., / Cornell University. Series "Working papers by Cornell University". 2017. No. 1702.00362.

This work is concerned with the optimal control problems governed by the 1D wave equation with variable coefficients and the control spaces $\mathcal M_T$ of either measure-valued functions $L^2(I,\mathcal M(\Omega))$
or vector measures $\mathcal M(\Omega,L^2(I))$. The cost functional involves the standard quadratic terms and the regularization term $\alpha\|u\|_{\mathcal M_T}$, $\alpha>0$. We construct and study three-level in time bilinear ...

Added: February 2, 2017

Romanov I., Alexey Shamaev, / Cornell University. Series math "arxiv.org". 2016. No. arXiv:1503.04461.

We will consider the exact controllability of the distributed system governed by wave equation with memory. It will be proved that this mechanical system can be driven to an equilibrium point in a finite time, the absolute value of the distributed control function being bounded. In this case the memory kernel is a linear combination ...

Added: March 21, 2015

Zlotnik A., Čiegis R., / Cornell University. Series arXiv "math". 2020. No. 2012.01000 [math.NA].

We study necessary conditions for stability of a Numerov-type compact higher-order finite-diffe\-rence scheme for the 1D homogeneous wave equation in the case of non-uniform spatial meshes. We first show that the uniform in time stability cannot be valid in any spatial norm provided that the complex eigenvalues appear in the associated mesh eigenvalue problem. Moreover, we prove ...

Added: December 3, 2020

Деркач М. М., Филимонов А. М., Filimonov D., М. : МИИТ, 2013

В настоящем выпуске помещены методические указания к решению задач по разделу «Метод бегущих волн» дисциплины «Уравнения математической физики». В первой части настоящих методических указаний метод бегущих волн рассматривается применительно к одномерному волновому уравнению. Во второй части рассматриваются гиперболические системы довольно общего вида, поскольку метод бегущих волн можно с успехом распространить и на гиперболические системы уравнений ...

Added: January 9, 2014

Romanov I., Shamaev A. S., Journal of Mathematical Sciences 2018 Vol. 234 No. 4 P. 470-484

We consider the problem of exact controllability for a system described by an equation with integral
“memory”. It is shown that, under certain conditions, this system can be brought to rest in finite time by
distributed control bounded in absolute value, and, in a special one-dimensional case, by control applied to
an end-point of the interval. We consider ...

Added: July 8, 2019

Petropavlovsky S., Цынков С. В., Журнал вычислительной математики и математической физики 2020 Т. 60 № 4 С. 725-737

A boundary method for simulation of wave propagation in 3D is proposed. The approach relies on the method of difference potentials and Huygens'principle allowing for an update of the solution only on the boundaries of a scatterer/ ...

Added: April 13, 2020

Romanov I., Shamaev A., Доклады Российской академии наук. Математика, информатика, процессы управления (ранее - Доклады Академии Наук. Математика) 2016 Т. 470 № 1 С. 22-25

The boundary control of vibrations of a plane membrane is considered. A constraint is imposed on the absolute value of the control function. The goal of the control is to drive the membrane to rest. Exact mathematical definitions are given. The proof technique used in this paper can be applied to a membrane of any ...

Added: September 8, 2016

Romanov I., Shamaev A., Journal of Dynamical and Control Systems 2013 Vol. 19 No. 4 P. 611-623

We will consider the exact controllability of the distributed system, governed by string equation with memory. It will be proved that this mechanical system can be driven to an equilibrium point in a finite time, the absolute value of the distributed control function being bounded. In this case, the memory kernel is a linear combination ...

Added: October 9, 2013