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1
Content available The existence of mild solutions and approximate controllability for nonlinear fractional neutral evolution systems
Echchaffani Zoubida, Aberqi Ahmed, Karite Touria, Leiva Hugo
International Journal of Applied Mathematics and Computer Science
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2024
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Vol. 34, no. 1
15--27
EN
The existence of mild solutions and approximate controllability for Riemann-Liouville fractional neutral evolution systems with nonlocal conditions of a fractional order is investigated. The Laplace transform and semigroup theory are the tools used to prove the existence. In turn, approximate controllability is proved on the basis of a Nemytskii operator, a Mittag-Leffler function and certain hypotheses using fixed point theorems, as well as the construction of a Cauchy sequence. An example is provided to highlight the main results.
2
Approximate controllability results in α-norm for some partial functional integrodifferential equations with nonlocal initial conditions in Banach spaces
Ndambomve Patrice, Kpoumie Moussa El-Khalil, Ezzinbi Khalil
Journal of Applied Analysis
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2023
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Vol. 29, nr 1
127--142
EN
In this work, we discuss the approximate controllability of some nonlinear partial functional integrodifferential equations with nonlocal initial condition in Hilbert spaces.We assume that the corresponding linear part is approximately controllable. The results are obtained by using fractional power theory and α-norm, the measure of noncompactness and theMönch fixed-point theorem, and the theory of analytic resolvent operators for integral equations. As a result, we obtain a generalization of the work of Mahmudov [N. I. Mahmudov, Approximate controllability of evolution systems with nonlocal conditions, Nonlinear Anal. 68 (2008), no. 3, 536-546], without assuming the compactness of the resolvent operator. Our results extend and complement many other important results in the literature. Finally, a concrete example is given to illustrate the application of the main results.
3
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.
4
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.
5
Content available On controllability of second order dynamical systems – a survey
Klamka J., Wyrwał J., Zawiski R.
Bulletin of the Polish Academy of Sciences. Technical Sciences
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2017
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Vol. 65, nr 3
279--295
EN
The paper presents a survey of recent results in the area of controllability of second order dynamical systems. Controllability problem for finite and infinite dimensional, linear, semilinear, deterministic and stochastic dynamical systems (with delays and undelayed) is taken into consideration. Different types of controllability are discussed.
6
Content available Schauder’s fixed-point theorem in approximate controllability problems
Babiarz A., Klamka J., Niezabitowski M.
International Journal of Applied Mathematics and Computer Science
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2016
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Vol. 26, no. 2
263--275
EN
The main objective of this article is to present the state of the art concerning approximate controllability of dynamic systems in infinite-dimensional spaces. The presented investigation focuses on obtaining sufficient conditions for approximate controllability of various types of dynamic systems using Schauder’s fixed-point theorem. We describe the results of approximate controllability for nonlinear impulsive neutral fuzzy stochastic differential equations with nonlocal conditions, impulsive neutral functional evolution integro-differential systems, stochastic impulsive systems with control-dependent coefficients, nonlinear impulsive differential systems, and evolution systems with nonlocal conditions and semilinear evolution equation.
7
Content available Approximate controllability of the impulsive semilinear heat equation
Leiva H., Merentes N.
Journal of Mathematics and Applications
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2015
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Vol. 38
85--104
EN
In this paper we apply Rothe's Fixed Point Theorem to prove the interior approximate controllability of the following semilinear impulsive Heat Equation [...] where k = 1, 2, . . . , p, Ω is a bounded domain in [...] is an open nonempty subset of Ω, 1ω denotes the characteristic function of the set ω, the distributed control u belongs to [...]. Under this condition we prove the following statement: For all open nonempty subsets ω of Ω the system is approximately controllable on [0, τ]. Moreover, we could exhibit a sequence of controls steering the nonlinear system from an initial state z0 to an ϵ neighborhood of the nal state z1 at time τ > 0.
8
Approximate controllability of semilinear impulsive strongly damped wave equation
Larez H., Leiva H., Rebaza J., Rios A.
Journal of Applied Analysis
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2015
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Vol. 21, nr 1
45--57
EN
Rothe’s fixed-point theorem is applied to prove the interior approximate controllability of a semilinear impulsive strongly damped wave equation with Dirichlet boundary conditions in the space Z1/2 = D((-Δ)1/2) × L2(Ω), where Ω is a bounded domain in Rn (n ≥ 1). Under some conditions we prove the following statement: For all open nonempty subsets ω of Ω the system is approximately controllable on [0, τ]. Moreover, we exhibit a sequence of controls steering the nonlinear system from an initial state z0 to a neighborhood of the final state z1 at time τ > 0.
9
Content available Existence and approximate controllability of Sobolev type fractional stochastic evolution equations
Mahmudov N. I.
Bulletin of the Polish Academy of Sciences. Technical Sciences
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2014
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Vol. 62, nr 2
205--215
EN
We study the existence of mild solutions and the approximate controllability concept for Sobolev type fractional semilinear stochastic evolution equations in Hilbert spaces. We prove existence of a mild solution and give sufficient conditions for the approximate controllability. In particular, we prove that the fractional linear stochastic system is approximately controllable in [0, b] if and only if the corresponding deterministic fractional linear system is approximately controllable in every [s, b], 0 ≤ s < b. An example is provided to illustrate the application of the obtained results.
10
Content available remote Dualities for linear control differential systems with infinite matrices
Mozyrska D., Bartosiewicz Z.
Control and Cybernetics
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2006
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Vol. 35, no 4
887-904
EN
Infinite-dimensional linear dynamic systems described by infinite matrices are studied. Approximate controllability for systems with lower-diagonal matrices is investigated, whereas observability is studied for systems with row-finite and upper-diagonal matrices. Different necessary or sufficient conditions of approximate controllability and observability of such systems are given. They are used to show dualities between these properties. The theorems on dualities extend the results known for finite-dimensional systems.
11
Content available remote Controllability and observability of distributed parameter systems : a survey
Klamka J.
Studia z Automatyki i Informatyki
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2002
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T. 27
7-19
EN
In this paper, various concepts of controllability and observability of linear infinite-dimensional continuous-time control systems with constant coefficients have been studied. The paper contains a survey of various fundamental definitions, theorems and corollaries concerning approximate controllability, approximate boundary controllability and observability. Moreover, the different relations among them are also listed. As an illustration of the general theory, constrained approximate controllability, constrained approximate boundary controllability and observability of linear distributed parameter dynamical systems described by partial differential state equations and linear output equation are considered.
PL
W pracy rozważono różne koncepcje sterowalności i obserwowalności liniowych, nieskończenie wymiarowych, ciągłych układów sterowania o stałych współczynnikach. Praca zawiera przegląd różnych podstawowych definicji, twierdzeń i wniosków dotyczących sterowalności aproksymacyjnej, sterowalności aproksymacyjnej brzegowej i obserwowalności. Ponadto przedstawiono różne związki między nimi. Jako ilustrację ogólnej teorii rozważono sterowalność aproksymacyjną z ograniczeniami, sterowalność aproksymacyjną brzegową z ograniczeniami oraz obserwowalność liniowych układów o parametrach rozłożonych, opisanych cząstkowymi równaniami różniczkowymi stanu i liniowym równaniem wyjścia.
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