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1
Content available Heating control of a finite rod with a mobile source
Jilavyan Samvel H., Grigoryan Edmon R., Khurshudyan Asatur Zh.
Archives of Control Sciences
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2021
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Vol. 31, No. 2
417--430
EN
The Green’s function approach is applied for studying the exact and approximate null-controllability 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.
2
Content available Approximate controllability of second order infinite dimensional systems
Klamka Jerzy, Khurshudyan Asatur Zh.
Archives of Control Sciences
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2021
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Vol. 31, No. 1
165--184
EN
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.
3
Content available Exact and approximate distributed controllability of processes described by KdV and Boussinesq equations: The Green’s function approach
Klamka Jerzy, Avetisyan Ara S., Khurshudyan Asatur Zh.
Archives of Control Sciences
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2020
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Vol. 30, No. 1
177--193
EN
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.
4
Content available Heuristic control of nonlinear power systems: Application to the infinite bus problem
Hossain Musabbir Md, Khurshudyan Asatur Zh.
Archives of Control Sciences
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2019
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Vol. 29, No. 2
279--288
EN
In this paper, we apply the heuristic method for determination of control functions for controllability analysis of nonlinear power systems. The problem of control of quasi-linear systems under proper assumptions on the nonlinear term is considered in the general statement. Making use of the Green’s function solution of nonlinear systems, the exact and approximate controllability conditions are expressed in terms of unknown controls in an explicit form. The way of resolving controls determination is discussed. As a particular application, a one-machine infinite-bus system is considered described by a coupled system of three first order ordinary differential equations. Two heuristic forms of admissible controls are considered providing approximate controllability within the same amount of time having different intensities. Results of numerical simulations are presented and discussed.
5
Content available Distributed controllability of one-dimensional heat equation in unbounded domains: The Green's function approach
Khurshudyan Asatur Zh.
Archives of Control Sciences
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2019
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Vol. 29, No. 1
57--71
EN
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.
6
Content available Averaged controllability of heat equation in unbounded domains with random geometry and location of controls: The Green’s function approach
Klamka Jerzy, Khurshudyan Asatur Zh.
Archives of Control Sciences
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2019
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Vol. 29, No. 4
573--584
EN
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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