Estimating Nonstationary Parameters of Differential Equations Under Uncertainty
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ESTIMATING NONSTATIONARY PARAMETERS OF DIFFERENTIAL EQUATIONS UNDER UNCERTAINTY O. G. Nakonechnyi,1† Iu. M. Shevchuk,1‡ and V. K. Chikrii1
UDC 517.9:517.87
Abstract. The algorithms of constructing optimal functional estimates and guaranteed estimates of nonstationary parameters of differential equations are proposed. The results are generalized to discrete-time models. These algorithms can be used to predict the dynamics of systems of differential equations. The results of a numerical experiment for the problem of constructing guaranteed estimates for the mathematical model of propagation of one type of information are considered. Keywords: differential equations, nonstationary parameters, optimal functional estimation, guaranteed estimation, uncertainty. Mathematical models under uncertainty generate a number of problems which are considered in particular in [1–3]. One of them is estimating system parameters whose construction algorithms are analyzed in [4, 5]. Let on the interval t Î(0, T ) , a vector function x ( t ) Î R n be observed, which is a generalized solution of the equation
x&( t ) = F ( t, x ( t )) j ( t ) + f ( t, x ( t )) + h ( t ) ,
(1)
where F ( t, x ( t )) is a given n ´ m matrix function, f ( t, x ( t )) Î R n is a given vector function, and j( t ) Î R m and
h( t ) Î R n are unknown vector functions. Denote by L2 , m (0, T ) and L2 , n (0, T ) spaces of measurable functions, square integrable on (0, T ) , from spaces R m and R n , respectively. Assume that F ( t, x ( t )) and f ( t, x ( t )) are bounded and continuous on (0, T ) functions of their arguments, function j( t ) ÎQ , t Î(0, T ) , where Q is the class of k – 1 ( k > 1) times continuously differentiable vector functions for which there exists a generalized derivative of k th order; assume also that j( t ) belongs to space L2 , m (0, T ) , and function h( t ) to space L2 , n (0, T ) . Definition 1. By the generalized solution of Eq. (1) with the initial condition x (0) = x 0 we understand vector function x ( t ) , which is the solution of the integral equation t
t
x ( t ) = x 0 + ò F ( t, x ( t)) j ( t) dt + ò f ( t, x ( t)) dt + h 1 ( t ), 0 < t £ T , 0
t
(2)
0
where h 1 ( t ) = ò h ( t) dt . 0
Assume that the solution of Eq. (2) exists and is unique. The problem is to find optimal (in a certain sense) estimate of function j( t ) ÎQ , t Î(0, T ) , with the given observations x ( t ) , t Î(0, T ) , and known constraints imposed on functions j( t ) and h 1 ( t ) . 1
Taras Shevchenko National University of Kyiv, Kyiv, Ukraine, †[email protected]; ‡[email protected]. Translated from Kibernetika i Sistemnyi Analiz, No. 4, July–August, 2018, pp. 109–121. Original article submitted February 26, 2018. 610
1060-0396/18/5404-0610 ©2018 Springer Science+Business Media, LLC
Assume that ( j, h 1 ) ÎG and set G is defined as
G = { ( j, h 1 ) : F( j, h 1 ) £ g 2 (T )}, T
T
F( j, h 1 ) = ò q12 ( t) | j ( k ) ( t) | 2 dt + ò q 22 ( t) | h 1 ( t) | 2 dt , 0
0
where q i ( t ), i = 1, 2 , are functions continuous on (0, T ) and
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