Search results

Filters

  • Journals
  • Authors
  • Keywords
  • Date
  • Type

Search results

Number of results: 3
items per page: 25 50 75
Sort by:
Download PDF Download RIS Download Bibtex

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.
Go to article

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.
Go to article

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
Download PDF Download RIS Download Bibtex

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.

Go to article

Authors and Affiliations

Asatur Zh. Khurshudyan
Download PDF Download RIS Download Bibtex

Abstract

In this paper we prove the exact controllability of a time varying semilinear system considering non-instantaneous impulses, delay, and nonlocal conditions occurring simultaneously. It is done by using the Rothe’s fixed point theorem together with some sub-linear conditions on the nonlinear term, the impulsive functions, and the function describing the nonlocal conditions. Furthermore, a control steering the semilinear system from an initial state to a final state is exhibited.
Go to article

Authors and Affiliations

Dalia Cabada
1
Katherine Garcia
2
Cristi Guevara
3
Hugo Leiva
2
ORCID: ORCID

  1. School of Mathematical and Statistical Sciences, Arizona State University, United States of America
  2. School of Mathematical Sciences and Information Technology, Yachay Tech University, Ecuador
  3. Arizona Department of Education, United States of America

This page uses 'cookies'. Learn more