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Abstract

This paper presents an overview of basic concepts, features and difficulties of the boundary element method (BEM) and examples of its application to exterior and interior problems. The basic concepts of the BEM are explained firstly, and different methods for treating the non-uniqueness problem are described. The application of the BEM to half-space problems is feasible by considering a Green's Function that satisfies the boundary condition on the infinite plane. As a special interior problem, the sound field in an ultrasonic homogenizer is computed. A combination of the BEM and the finite element method (FEM) for treating the problem of acoustic-structure interaction is also described. Finally, variants of the BEM are presented, which can be applied to problems arising in flow acoustics.
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Authors and Affiliations

Rafael Piscoya
Martin Ochmann
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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

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

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

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

In recent years, type-II superlattice-based devices have completed the offer of the electronic industry in many areas of applications. Photodetection is one of them, especially in the mid-infrared wavelength range. It is due to the unique feature of a superlattice material, which is a tuneable bandgap. It is also believed that the dark current of superlattice-based photodetectors is strongly suppressed due to the suppression of the band-to-band tunnelling current in a superlattice material. This argument relies, however, on a semi-classical approach that treats superlattice as a bulk material with effective parameters extracted from the kp analysis. In the paper, a superlattice device is analysed on a quantum level: the non-equilibrium Green’s function method is applied to the two-band Hamiltonian of the InAs/GaSb superlattice p-i-n diode. The analysis concentrates on the band-to-band tunnelling with the aim to validate the correctness of a semi-classical description of the phenomenon. The results of calculations reveal that in a superlattice diode, the inter-band tunnelling occurs only for certain values of energy and in-plane momentum, for which electronic and hole sub-bands cross. The transitions occurring for vanishing in-plane momentum produce resonances in the current-voltage characteristics – the feature which was reported in a few experimental observations. This scenario is quite different from that occurring in bulk materials, where there is a range of energy-momentum pairs for which the band-to-band tunnelling takes place, and so current-voltage characteristics are free from any resonances. However, simulations show that, while not justified for a detailed analysis, the semi-classical description can be applied to superlattice-based devices for an ‘order of magnitude’ estimation of the band-to-band tunnelling current.
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Authors and Affiliations

Marcin Makowiec
1
ORCID: ORCID
Andrzej Kolek
1
ORCID: ORCID

  1. Department of Electronics Fundamentals, Rzeszow University of Technology, al. Powstańców Warszawy 12, Rzeszów 35-959, Poland

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