Problems of Control of the Dynamics of Incompletely Defined Three-Dimensional Elastic Bodies. II. Discretely Defined Des
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PROBLEMS OF CONTROL OF THE DYNAMICS OF INCOMPLETELY DEFINED THREE-DIMENSIONAL ELASTIC BODIES. II. DISCRETELY DEFINED DESIRED STATE
V. A. Stoyan
UDC 517.95:519.86:539.3
Abstract. The author solves the problems of control of linearly transformed vector function of displacement points of three-dimensional elastic body with the purpose of root-mean-square approximation to its discretely defined values. The problems are solved without constraints on the body geometry and under discretely defined observations of its initial-boundary state. Spatially, superficially, and initially distributed external-dynamic perturbations are considered as control factors. The evaluation of accuracy and uniqueness of control is conducted. Keywords: spatially distributed dynamic systems, spatial problems of elastic theory, pseudoinversion, control.
INTRODUCTION The present paper continues author’s studies [1] in the dynamics of three-dimensional elastic bodies that operate under incomplete information about their initial-superficial state. The studies [2, 3] solve problems of constructing mathematical models of the dynamics of such bodies that are degenerate in one spatial coordinate for the cases where discretely and continuously defined observations are carried out for the initial state of interior points and current state of superficial points of the body and the constructed mathematical model is coordinated with them in the root-mean-square criterion. The papers [4, 5] solve problems of control of the considered bodies based on the root-mean-square approximation of their state to a predetermined state. These papers consider cases of control of spatially, superficially, and initially distributed external-dynamic perturbations taken by one, two, and three. The paper [1] solves these problems for three-dimensional elastic bodies with arbitrary geometry of their surface. And continuously defined desired state and continuously defined observations of boundary-superficial state of the body are taken into account based on the root-mean-square criterion. These are new problems of the mechanics of deformable solids, not solved yet and being of practical importance. Even more complicated are dynamic control problems for the case where information about initial-superficial and desired states of the bodies is defined discretely and control is performed by both discretely and continuously defined external-dynamic control factors: initial, superficial, and spatial. The present publication solves these problems. We will obtain analytical expressions for both external-dynamic controls and the field of dynamic displacements of points of a body coordinated in quadratic mean with the desired one and will evaluate the accuracy of the obtained solutions and analyze the conditions of their uniqueness.
Taras Shevchenko National University of Kyiv, Kyiv, Ukraine, [email protected]. Translated from Kibernetika i Sistemnyi Analiz, No. 5, September–October, 2017, pp. 102–112. Original article submitted December 6, 2016. 1060-0396/17/5305-0743 ©2017 Springe
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