Infinite system of 2-systems of differential equations in Hilbert space Ibragimov G. I., Qushaqov H. Sh. Abstract

0.1 Introduction

Some control problems represented by partial differential equations (PDE) can be reduced to control problems for an infinite system of differential equations by the decomposition method which is one of the major methods to solve such problems(see, for example, [6, 1, 20, 2, 5, 4, 21]).

Therefore there is a significant relationship between the control problems for the partial differential equations and those for the infinite system of differential equations. For instance, the studies [14, 18, 17, 20] reduced a differential game problem that is given by a linear partial differential equation of the form

where and are the control parameters of pursuer and evader respectively, is a scalar function, to a differential game represented by the following infinite system of differential equations

(0.1.1)

where and , , are control parameters of pursuer and evader respectively, , and coefficients , , satisfy the condition .

The works [18, 17, 20] suggest that we should study differential game problems for the infinite system of differential equations (0.1.1) in one theoretical frame independently of partial differential equations assuming that , , in (0.1.1) are any real numbers.

We recall the vector space of all sequences of real numbers

is a Hilbert space with the inner product and norm defined by

Existence and uniqueness theorem for the infinite system (0.1.1) was proved in [13] for any positive numbers , , in a Hilbert space connected with the operator . Further, in [8] existence and uniqueness theorem for the infinite system of differential equations

(0.1.2)

was proved in Hilbert space , where are real numbers, , the function , , whose components are measurable and satisfy the condition

(0.1.3)

is a sufficiently large fixed number. The main purpose of the present work is to investigate the following infinite system of differential equations

(0.1.4)

in Hilbert space , where is a given non negative real numbers,

The class of functions with measurable coordinates , , , satisfying the condition

(0.1.5)

we denote by , where is a given positive number.

The problem is to determine if there exist a unique solution of the infinite system of differential equations (0.1.4) in the Hilbert space ?