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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.

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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.

3

An output controllability problem for semilinear distributed hyperbolic systems

Zerrik E., Larhrissi R., Bourray H.

International Journal of Applied Mathematics and Computer Science

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2007

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Vol. 17, no 4

437-446

EN

The paper aims at extending the notion of regional controllability developed for linear systems to the semilinear hyperbolic case. We begin with an asymptotically linear system and the approach is based on an extension of the Hilbert uniqueness method and Schauder’s fixed point theorem. The analytical case is then tackled using generalized inverse techniques and converted to a fixed point problem leading to an algorithm which is successfully implemented numerically and illustrated with examples.

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On the Constrained Controllability of Dynamical Systems With Multiple Delays in the State

Sikora B.

International Journal of Applied Mathematics and Computer Science

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2003

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Vol. 13, no 4

469-479

EN

Linear stationary dynamical systems with multiple constant delays in the state are studied. Their relative and approximate controllability properties with constrained controls are discussed. Definitions of various types of controllability with constrained controls for systems with delays in the state are introduced. Some theorems concerning the relative and the approximate relative controllability with constrained controls for dynamical systems with delays in the state are established. Various types of constraints are considered. Numerical examples illustrate the theoretical analysis. An example of a real technical dynamical system is given to indicate one of possible practical applications of the theoretical results.

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Constrained Controllability of Dynamic Systems

Klamka J.

International Journal of Applied Mathematics and Computer Science

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1999

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Vol. 9, no 2

231-244

EN

The present paper is devoted to a study of constrained controllability and controllability for linear dynamic systems if the controls are taken to be non-negative. By analogy to the usual definition of controllability it is possible to introduce the concept of positive controllability. Weshall concentrate on approximate positive controllability for linear infinite-dimensional dynamic systems when the values of controls are taken from a positive closed convex cone and the operator of the system is normal and has pure discrete point spectrum. Special attention is paid to positive infinite-dimensional linear dynamic systems. General approximate constrained controllability results are then applied to distributed-parameter dynamic systems described by linear partial-differential equations of parabolic type with various kinds of boundary conditions. Several remarks and comments on the relationships between different concepts of controllability are given. Finally, a simple illustrative example is also presented.