Identification of the Functions of Response to Loading for Stationary Systems
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IDENTIFICATION OF THE FUNCTIONS OF RESPONSE TO LOADING FOR STATIONARY SYSTEMS
UDC 519.626.6
V. M. Abdullayev
Abstract. We investigate the solution to the parametric identification problem for loaded systems of differential equations. We propose to use iterative methods based on the first-order optimization methods. For this purpose, we obtain formulas for the gradient of the objective functional, which assesses the adequacy degree of the obtained parameters. The results of numerical solution to some test problems are given. Keywords: loaded differential equations, response to loading, optimal control, nonlocal conditions, inverse problem. INTRODUCTION Processes in ecology, oil and gas filtration, motion of ground water, and many others are known to be described by systems of loaded ordinary or partial differential equations [1–4]. The loading points, more exactly the states at these points, influence (as well/or) the state at all points of the object (process). In this connection, optimization of response functions is important for efficient operation of the object. Inverse problems with respect to loaded equations where it is required to identify response functions often occur in practice. To this end, additional information about the object is necessary; it can be specified in the form of point, integral, nondivided point values of the state of the process [5–8]. One of the approaches to solution of the inverse problem is reducing it to the optimal control problem with respect to functions of response to loading with the use of the functional of root mean square deviation of desired (observed) states of the process from those calculated by means of the mathematical model. In the present paper, we will propose a technique to solve the problem of optimizing functions of response to loading with application of first-order optimal control methods. To this end, we derive formulas for the gradient of the functional with respect to the parameters being optimized. We will use the solution of test problems as an example to prove the efficiency of the proposed approach to the numerical solution of the identification problem. PROBLEM STATEMENT Consider the following system of loaded differential equations [4]: l
du( x ) = A ( x ) u( x ) + å B s ( x ) u( x s ) + C ( x ), x Î[ a, b], dx s=1
(1)
Azerbaijan State Oil and Industry University; Institute of Control Systems of the National Academy of Sciences of Azerbaijan, Baku, Azerbaijan, [email protected]. Translated from Kibernetika i Sistemnyi Analiz, No. 3, May–June, 2017, pp. 100–110. Original article submitted June 15, 2016. 1060-0396/17/5303-0417 ©2017 Springer Science+Business Media New York
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where A ( x ) = ( Aij ( x )) and B s ( x ) = ( B ijs ) , x Î[ a, b] , are n-dimensional quadratic matrix continuous functions; C ( x ) is a continuous n -dimensional vector function; u( x ) Î R n is state of the object at point x Î[ a, b]; x s Î[ a, b], s = 1, 2, K , l , are given places of loading. We will call matrix functions B ( x ) = ( B 1 ( x ), B 2 ( x ), K , B
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