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Abstract

We derive exact and approximate controllability conditions for the linear one-dimensional heat equation in an infinite and a semi-infinite domains. The control is carried out by means of the time-dependent intensity of a point heat source localized at an internal (finite) point of the domain. By the Green’s function approach and the method of heuristic determination of resolving controls, exact controllability analysis is reduced to an infinite system of linear algebraic equations, the regularity of which is sufficient for the existence of exactly resolvable controls. In the case of a semi-infinite domain, as the source approaches the boundary, a lack of L2-null-controllability occurs, which is observed earlier by Micu and Zuazua. On the other hand, in the case of infinite domain, sufficient conditions for the regularity of the reduced infinite system of equations are derived in terms of control time, initial and terminal temperatures. A sufficient condition on the control time, heat source concentration point and initial and terminal temperatures is derived for the existence of approximately resolving controls. In the particular case of a semi-infinite domain when the heat source approaches the boundary, a sufficient condition on the control time and initial temperature providing approximate controllability with required precision is derived.

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Authors and Affiliations

Asatur Zh. Khurshudyan
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Abstract

We study the exact and approximate controllabilities of the Langevin equation describing the Brownian motion of particles with a white noise. The Langevin equation is shown to describe also the bacterial run-and-tumble motion. Applying the Green’s function approach to the Green’s function representation of the Langevin equation, we obtain necessary and sufficient conditions for exact controllability in the form of a finite-dimensional problem of moments. For the approximate controllability, we obtain only sufficient conditions. The sets of resolving controls are characterized in both cases. The theoretical derivations are supported by a numerical analysis.

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Authors and Affiliations

Asatur Zh. Khurshudyan
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Abstract

Abstract We propose a generalization of the Butkovskiy's method of control with compact support [1] allowing to derive exact controllability conditions and construct explicit solutions in control problems for systems with distributed parameters. The idea is the introduction of a new state function which is supported in considered bounded time interval and coincides with the original one therein. By means of techniques of the distributions theory the problem is reduced to an interpolation problem for Fourier image of unknown function or to corresponding system of integral equalities. Treating it as infinite dimensional problem of moments, its L1, L2 and L∞-optimal solutions are constructed explicitly. The technique is explained for semilinear wave equation with distributed and boundary controls. Particular cases are discussed.
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Authors and Affiliations

Asatur Zh. Khurshudyan
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Abstract

The constrained averaged controllability of linear one-dimensional heat equation defined on R and R+ is studied. The control is carried out by means of the time-dependent intensity of a heat source located at an uncertain interval of the corresponding domain, the end-points of which are considered as uniformly distributed random variables. Employing the Green’s function approach, it is shown that the heat equation is not constrained averaged controllable neither in R nor in R+. Sufficient conditions on initial and terminal data for the averaged exact and approximate controllabilities are obtained. However, constrained averaged controllability of the heat equation is established in the case of point heat source, the location of which is considered as a uniformly distributed random variable. Moreover, it is obtained that the lack of averaged controllability occurs for random variables with arbitrary symmetric density function.

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Authors and Affiliations

Jerzy Klamka
Asatur Zh. Khurshudyan
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Abstract

The Bulletin of the Polish Academy of Sciences: Technical Sciences (Bull.Pol. Ac.: Tech.) is published bimonthly by the Division IV Engineering Sciences of the Polish Academy of Sciences, since the beginning of the existence of the PAS in 1952. The journal is peer‐reviewed and is published both in printed and electronic form. It is established for the publication of original high quality papers from multidisciplinary Engineering sciences with the following topics preferred: Artificial and Computational Intelligence, Biomedical Engineering and Biotechnology, Civil Engineering, Control, Informatics and Robotics, Electronics, Telecommunication and Optoelectronics, Mechanical and Aeronautical Engineering, Thermodynamics, Material Science and Nanotechnology, Power Systems and Power Electronics.

Journal Metrics: JCR Impact Factor 2018: 1.361, 5 Year Impact Factor: 1.323, SCImago Journal Rank (SJR) 2017: 0.319, Source Normalized Impact per Paper (SNIP) 2017: 1.005, CiteScore 2017: 1.27, The Polish Ministry of Science and Higher Education 2017: 25 points.

Abbreviations/Acronym: Journal citation: Bull. Pol. Ac.: Tech., ISO: Bull. Pol. Acad. Sci.-Tech. Sci., JCR Abbrev: B POL ACAD SCI-TECH Acronym in the Editorial System: BPASTS.

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Authors and Affiliations

Md Musabbir Hossain
Asatur Zh. Khurshudyan
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Abstract

Abstract Distribution optimization of elastic material under elastic isotropic rectangular thin plate subjected to concentrated moving load is investigated in the present paper. The aim of optimization is to damp its vibrations in finite (fixed) time. Accepting Kirchhoff hypothesis with respect to the plate and Winkler hypothesis with respect to the base, the mathematical model of the problem is constructed as two-dimensional bilinear equation, i.e. linear in state and control function. The maximal quantity of the base material is taken as optimality criterion to be minimized. The Fourier distributional transform and the Bubnov-Galerkin procedures are used to reduce the problem to integral equality type constraints. The explicit solution in terms of two- dimensional Heaviside‘s function is obtained, describing piecewise-continuous distribution of the material. The determination of the switching points is reduced to a problem of nonlinear programming. Data from numerical analysis are presented.
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Authors and Affiliations

Samvel H. Jilavyan
Asatur Zh. Khurshudyan
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Abstract

The Green’s function approach is applied for studying the exact and approximate nullcontrollability of a finite rod in finite time by means of a source moving along the rod with controllable trajectory. The intensity of the source remains constant. Applying the recently developed Green’s function approach, the analysis of the exact null-controllability is reduced to an infinite system of nonlinear constraints with respect to the control function. A sufficient condition for the approximate null-controllability of the rod is obtained. Since the exact solution of the system of constraints is a long-standing open problem, some heuristic solutions are used instead. The efficiency of these solutions is shown on particular cases of approximate controllability.
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Bibliography

[1] J. Klamka: Controllability of Dynamical Systems. Kluwer Academic, Dordrecht, 1991.
[2] S.A. Avdonin and S.A. Ivanov: Families of Exponentials. The Method of Moments in Controllability Problems for Distributed Parameter Systems. Cambridge University Press, New York, 1995.
[3] A. Fursikov and O.Yu. Imanuvilov: Controllability of Evolution Equations. Lecture Notes Series, vol. 34. Seoul National University, Research Institute of Mathematics, Global Analysis Research Center, Seoul, 1996.
[4] E. Zuazua: Controllability and Observability of Partial Differential Equations: Some Results and Open Problems. Handbook of Differential Equations: Evolutionary Differential Equations, vol. 3, Elsevier/North-Holland, Amsterdam, 2006.
[5] R. Glowinski, J.-L. Lions and J. He: Exact and Approximate Controllability for Distributed Parameter Systems: A Numerical Approach. Cambridge University Press, New York, 2008.
[6] A.S. Avetisyan and As.Zh. Khurshudyan: Controllability of Dynamic Systems: The Green’s Function Approach. Cambridge Scholars Publishing, Cambridge, 2018.
[7] S. Micu and E. Zuazua: On the lack of null-controllability of the heat equation on the half-line. Transactions of the American Mathematical Society, 353(4), (2001), 1635–1659.
[8] S. Micu and E. Zuazua: Null Controllability of the Heat Equation in Unbounded Domains. In “Unsolved Problems in Mathematical Systems and Control Theory”, edited by Blondel V.D., Megretski A., Princeton University Press, Princeton, 2004.
[9] V. Barbu: Exact null internal controllability for the heat equation on unbounded convex domain. ESAIM: Control, Optimisation and Calculus of Variations, 20 (2014), 222–235, DOI: 10.1051/cocv/2013062.
[10] As.Zh. Khurshudyan: (2019), Distributed controllability of heat equation in un-bounded domains: The Green’s function approach. Archives of Control Sciences, 29(1), (2019), 57–71, DOI: 10.24425/acs.2019.127523.
[11] S. Ivanov and L. Pandolfi: Heat equation with memory: Lack of controllability to rest. Journal of Mathematical Analysis and Applications, 355 (2009), 1–11, DOI: 10.1016/j.jmaa.2009.01.008.
[12] A. Halanay and L. Pandolfi: Approximate controllability and lack of controllability to zero of the heat equation with memory. Journal of Mathematical Analysis and Applications, 425 (2015), 194–211, DOI: 10.1016/j.jmaa.2014.12.021.
[13] B.S. Yilbas: Laser Heating Applications: Analytical Modelling. Elsevier, Waltham, 2012.
[14] A.G. Butkovskiy and L.M. Pustylnikov: Mobile Control of Distributed Parameter Systems. Chichester, Ellis Horwood, 1987.
[15] V.A. Kubyshkin and V.I. Finyagina: Moving control of systems with distributed parameters (in Russian). Moscow: SINTEG, 2005.
[16] Sh.Kh. Arakelyan and As.Zh. Khurshudyan: The Bubnov-Galerkin procedure for solving mobile control problems for systems with distributed parameters. Mechanics. PNAS Armenia, 68(3), (2015), 54–75.
[17] A.G. Butkovskiy: Some problems of control of the distributed-parameter systems. Automation and Remote Control, 72 (2011), 1237–1241, DOI: 10.1134/S0005117911060105.
[18] A.S. Avetisyan and As.Zh. Khurshudyan: Green’s function approach in approximate controllability problems. Proceedings of National Academy of Sciences of Armenia. Mechanics, vol. 69, issue 2, (2016), 3–22, DOI: 10.33018/69.2.1.
[19] A.S. Avetisyan and As.Zh. Khurshudyan: Green’s function approach in approximate controllability of nonlinear physical processes. Modern Physics Letters A, 32 1730015, (2017), DOI: 10.1142/S0217732317300154.
[20] As.Zh. Khurshudyan: Resolving controls for the exact and approximate controllabilities of the viscous Burgers’ equation: the Green’s function approach. International Journal of Modern Physics C, 29(6), 1850045, (2018), DOI: 10.1142/S0129183118500456.
[21] A.S. Avetisyan and As.Zh. Khurshudyan: Exact and approximate controllability of nonlinear dynamic systems in infinite time: The Green’s function approach. ZAMM, 98(11), (2018), 1992–2009, DOI: 10.1002/zamm.201800122.
[22] As.Zh. Khurshudyan: Exact and approximate controllability conditions for the micro-swimmers deflection governed by electric field on a plane: The Green’s function approach. Archives of Control Sciences, 28(3), (2018), 335–347. DOI: 10.24425/acs.2018.124706.
[23] J. Klamka and As.Zh. Khurshudyan: Averaged controllability of heat equation in unbounded domains with uncertain geometry and location of controls: The Green’s function approach. Archives of Control Sciences, 29(4), (2019), 573–584, DOI: 10.24425/acs.2018.124706.
[24] J. Klamka, A.S. Avetisyan and As.Zh. Khurshudyan: Exact and approximate distributed controllability of the KdV and Boussinesq equations: The Green’s function approach. Archives of Control Sciences, 30(1), (2020), 177–193, DOI: 10.24425/acs.2020.132591.
[25] J. Klamka and As.Zh. Khurshudyan: Approximate controllability of second order infinite dimensional systems. Archives of Control Sciences, 31(1), (2021), 165–184, DOI: 10.24425/acs.2021.136885.
[26] As.Zh. Khurshudyan: Heuristic determination of resolving controls for exact and approximate controllability of nonlinear dynamic systems. Mathematical Problems in Engineering, (2018), Article ID 9496371, DOI: 10.1155/2018/9496371.
[27] H. Hossain and As.Zh. Khurshudyan: Heuristic control of nonlinear power systems: Application to the infinite bus problem. Archives of Control Sciences, 29(2), (2019), 279–288, DOI: 10.24425/acs.2019.129382.
[28] A.G. Butkovskii and L.M. Pustyl’nikov: Characteristics of Distributed- Parameter Systems. Kluwer Academic Publishers, 1993.
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Authors and Affiliations

Samvel H. Jilavyan
1
Edmon R. Grigoryan
1
Asatur Zh. Khurshudyan
2

  1. Faculty of Mathematics and Mechanics, Yerevan State University, 1 Alex Manoogian, 0025 Yerevan, Armenia
  2. Dynamicsof Deformable Systems and Coupled Fields, Institute of Mechanics, National Academy of Sciences of Armenia, 0019 Yerevan, Armenia
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Abstract

In the paper approximate controllability of second order infinite dimensional system with damping is considered. Applying linear operators in Hilbert spaces general mathematical model of second order dynamical systems with damping is presented. Next, using functional analysis methods and concepts, specially spectral methods and theory of unbounded linear operators, necessary and sufficient conditions for approximate controllability are formulated and proved. General result may be used in approximate controllability verification of second order dynamical system using known conditions for approximate controllability of first order system. As illustrative example using Green function approach approximate controllability of distributed dynamical system is also discussed.
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Authors and Affiliations

Jerzy Klamka
1
ORCID: ORCID
Asatur Zh. Khurshudyan
2

  1. Department of Measurements and Control Systems, Silesian University of Technology, Gliwice, Poland
  2. Institute of Mechanics, NAS of Armenia
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Abstract

In this paper, we study the constrained exact and approximate controllability of traveling wave solutions of Korteweg-de Vries (third order) and Boussinesq (fourth order) semi-linear equations using the Green’s function approach. Control is carried out by a moving external source. Representing the general solution of those equations in terms of the Frasca’s short time expansion, system of constraints on the distributed control is derived for both types of controllability. Due to the possibility of explicit solution provided by the heuristic method, the controllability analysis becomes straightforward. Numerical analysis confirms theoretical derivations.

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Authors and Affiliations

Jerzy Klamka
ORCID: ORCID
Ara S. Avetisyan
Asatur Zh. Khurshudyan

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