Bounded Solutions of a System of Linear Inhomogeneous Differential Equations of the First Order with Rectangular Matrice
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BOUNDED SOLUTIONS OF A SYSTEM OF LINEAR INHOMOGENEOUS DIFFERENTIAL EQUATIONS OF THE FIRST ORDER WITH RECTANGULAR MATRICES A. A. Boichuk1,2 and M. A. Elishevich3
UDC 517.926.7
We establish existence conditions and construct bounded solutions of a system of linear inhomogeneous differential equations of the first order with rectangular matrices.
Statement of the Problem In the present paper, we consider a system B(t)
dx = A(t)x + f (t), dt
t 2 R,
(1)
where A(t) and B(t) are m ⇥ n rectangular matrix functions and f (t) is a vector function of dimension m. Moreover, A(t), B(t), and f (t) are real and bounded functions with bounded derivatives of all orders for t 2 R. For this system, we determine the conditions of existence of its bounded solutions and construct these solutions under the indicated conditions. Main Definitions We use Jordan collections of vectors of the matrix B(t) with respect to the operator L(t) = A(t) − B(t)
d dt
and of the adjoint matrix B ⇤ (t) with respect to the operator L⇤ (t) = A⇤ (t) +
d ⇤ B (t) dt
formally adjoint to L(t). Definition 1 [1, p. 54]. At a point t 2 R, an element '(1) (t) 2 ker B(t) possesses a finite Jordan chain of vectors of the matrix B(t) with respect to the operator L(t) of length p, p ≥ 1, if there exist vectors '(i) (t), i = 1, p, satisfying the relations B(t)'(1) (t) = 0, 1
Institute of Mathematics, Ukrainian National Academy of Sciences, Kyiv, Ukraine; e-mail: [email protected]. Corresponding author. 3 Kiev National University of Building and Architecture, Kiev, Ukraine; e-mail: [email protected]. 2
Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 72, No. 6, pp. 758–775, June, 2020. Ukrainian DOI: 10.37863/umzh.v72i6.1059. Original article submitted September 4, 2019. 880
0041-5995/20/7206–0880
c 2020
Springer Science+Business Media, LLC
B OUNDED S OLUTIONS OF A S YSTEM OF L INEAR I NHOMOGENEOUS D IFFERENTIAL E QUATIONS OF THE F IRST O RDER
B(t)'(i) (t) = L(t)'(i−1) (t),
881
i = 2, p,
L(t)'(p) (t) 2 / Im B(t). Definition 2. At a point t 2 R, an element '˜(1) (t) 2 ker B(t) possesses a finite Jordan chain of vectors of the matrix B(t) with respect to the operator L(t) of length p˜, p˜ ≥ 1, if there exist vectors '˜(i) (t), i = 1, p˜, such that B(t)'˜(1) (t) = 0, B(t)'˜(i) (t) = L(t)'˜(i−1) (t),
i = 2, p˜,
L(t)'˜(˜p) (t) = 0. Definition 3. At a point t 2 R, an element 'ˆ(1) (t) possesses an auxiliary chain of vectors of the matrix B(t) with respect to the operator L(t) of length pˆ, pˆ ≥ 1, if there exist vectors 'ˆ(i) (t), i = 1, pˆ, satisfying the relations B(t)'ˆ(i) (t) = L(t)'ˆ(i−1) (t),
i = 2, pˆ,
B(t)'ˆ(1) (t) 2 / Im L(t), L(t)'ˆ(ˆp) (t) 2 / Im B(t). Similarly, we define chains of vectors on R. In what follows, unless otherwise specified, we assume that the established assertions are true on R. The properties of these chains on a finite segment of the real axis were studied in [1, pp. 54–57] (for square matrices and finite chains) and in [2, 3] (for rectangular matrices and finite, cyclic, and auxiliary chains). In [4, pp. 243–252],
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