Oscillation Criteria for Matrix Differential Equations

1967 ◽  
Vol 19 ◽  
pp. 184-199 ◽  
Author(s):  
H. C. Howard
Keyword(s):  
Existence Theorems ◽  
Symmetric Matrix ◽  
Order System ◽  
Matrix Differential Equation ◽  
First Order ◽  
The Matrix ◽  
Matrix Differential Equations ◽  
Matrix Differential ◽  
Image Position

We shall be concerned at first with some properties of the solutions of the matrix differential equation1.1whereis an n × n symmetric matrix whose elements are continuous real-valued functions for 0 < x < ∞, and Y(x) = (yij(x)), Y″(x) = (y″ ij(x)) are n × n matrices. It is clear such equations possess solutions for 0 < x < ∞, since one can reduce them to a first-order system and then apply known existence theorems (6, Chapter 1).

scholarly journals Perturbation analysis of a matrix differential equation ẋ = ABx

2018 ◽  
Vol 3 (1) ◽  
pp. 97-104 ◽  
Author(s):  
M. Isabel García-Planas ◽  
Tetiana Klymchuk
Keyword(s):  
Normal Form ◽  
Equivalence Transformations ◽  
First Order ◽  
Similarity Transformations ◽  
Canonical Pair ◽  
The Matrix ◽  
Matrix Differential Equations ◽  
Matrix Differential ◽  
Contragredient Equivalence ◽  
Canonical Matrix

AbstractTwo complex matrix pairs (A, B) and (A′, B′) are contragrediently equivalent if there are nonsingular S and R such that (A′, B′) = (S−1AR, R−1BS). M.I. García-Planas and V.V. Sergeichuk (1999) constructed a miniversal deformation of a canonical pair (A, B) for contragredient equivalence; that is, a simple normal form to which all matrix pairs (A + A͠, B + B͠) close to (A, B) can be reduced by contragredient equivalence transformations that smoothly depend on the entries of A͠ and B͠. Each perturbation (A͠, B͠) of (A, B) defines the first order induced perturbation AB͠ + A͠B of the matrix AB, which is the first order summand in the product (A + A͠)(B + B͠) = AB + AB͠ + A͠B + A͠B͠. We find all canonical matrix pairs (A, B), for which the first order induced perturbations AB͠ + A͠B are nonzero for all nonzero perturbations in the normal form of García-Planas and Sergeichuk. This problem arises in the theory of matrix differential equations ẋ = Cx, whose product of two matrices: C = AB; using the substitution x = Sy, one can reduce C by similarity transformations S−1CS and (A, B) by contragredient equivalence transformations (S−1AR, R−1BS).

On matrix differential equations with several unbounded delays

2006 ◽  
Vol 17 (4) ◽  
pp. 417-433 ◽  
Author(s):  
J. ĈERMÁK
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Delay Equation ◽  
Matrix Differential Equation ◽  
Continuous Vector ◽  
Asymptotic Bounds ◽  
The Matrix ◽  
Exponentially Stable ◽  
Matrix Differential Equations ◽  
Matrix Differential

The paper focuses on the matrix differential equation \[ \dot y(t)=A(t)y(t)+\sum_{j=1}^{m}B_j(t)y(\tau_j(t))+f(t),\quad t\in I=[t_0,\infty)\vspace*{-3pt} \] with continuous matrices $A$, $B_j$, a continuous vector $f$ and continuous delays $\tau_j$ satisfying $\tau_k\circ\tau_l =\tau_l\circ\tau_k$ on $I$ for any pair $\tau_k,\tau_l$. Assuming that the equation \[ \dot y(t)=A(t)y(t)\] is uniformly exponentially stable, we present some asymptotic bounds of solutions $y$ of the considered delay equation. A system of simultaneous Schröder equations is used to formulate these asymptotic bounds.

Existence Theorems for Nonlinear Boundary Value Problems

1974 ◽  
Vol 26 (4) ◽  
pp. 884-892 ◽  
Author(s):  
W. L. McCandless
Keyword(s):  
Nonlinear Boundary ◽  
Existence Theorems ◽  
Boundary Value ◽  
Continuous Functions ◽  
Order System ◽  
Nonlinear Boundary Value Problems ◽  
First Order ◽  
Differentiable Functions ◽  
Image Position ◽  
General Boundary Value Problem

Let C(I) denote the linear space of continuous functions from the compact interval I = [a, b] into n-dimensional real arithmetic space Rn, and let C′(I) be the subspace of continuously differentiable functions on I. A general boundary value problem for a first-order system of n ordinary differential equations on I is given by

A Converse Problem in Matrix Differential Equations

1973 ◽  
Vol 16 (3) ◽  
pp. 401-403
Author(s):  
Warren E. Shreveo
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Matrix Differential Equation ◽  
Converse Problem ◽  
Matrix Differential Equations ◽  
Matrix Differential ◽  
Image Position

Suppose X and F are nxn matrix solutions of the n X n matrix differential equation(1)such that(2)where J is some interval.

scholarly journals ON CONDITIONING FOR A GENERAL SYSTEM OF FOUR POINT BOUNDARY VALUE PROBLEMS

1996 ◽  
Vol 27 (3) ◽  
pp. 219-225
Author(s):  
M. S. N. MURTY
Keyword(s):  
Differential Equation ◽  
Fundamental Matrix ◽  
Close Relationships ◽  
General System ◽  
Point Boundary ◽  
Matrix Differential Equation ◽  
Growth Behaviour ◽  
First Order ◽  
Matrix Differential ◽  
The Stability

In this paper we investigate the close relationships between the stability constants and the growth behaviour of the fundamental matrix to the general FPBVP'S associated with the general first order matrix differential equation.

scholarly journals Explicit solutions for a system of coupled Lyapunov differential matrix equations

1987 ◽  
Vol 30 (3) ◽  
pp. 427-434 ◽  
Author(s):  
L. Jodar ◽  
M. Mariton
Keyword(s):  
Cauchy Problem ◽  
Explicit Expression ◽  
Linear Quadratic ◽  
The Cauchy Problem ◽  
Matrix Differential Equations ◽  
Lyapunov Matrix ◽  
Error Accumulation ◽  
Matrix Differential ◽  
Image Position ◽  
Differential Matrix

This paper is concerned with the problem of obtaining explicit expressions of solutions of a system of coupled Lyapunov matrix differential equations of the typewhere Fi, Ai(t), Bi(t), Ci(t) and Dij(t) are m×m complex matrices (members of ℂm×m), for 1≦i, j≦N, and t in the interval [a,b]. When the coefficient matrices of (1.1) are timeinvariant, Dij are scalar multiples of the identity matrix of the type Dij=dijI, where dij are real positive numbers, for 1≦i, j≦N Ci, is the transposed matrix of Bi and Fi = 0, for 1≦i≦N, the Cauchy problem (1.1) arises in control theory of continuous-time jump linear quadratic systems [9–11]. Algorithms for solving the above particular case can be found in [12]]. These methods yield approximations to the solution. Without knowing the explicit expression of the solutions and in order to avoid the error accumulation it is interesting to know an explicit expression for the exact solution. In Section 2, we obtain an explicit expression of the solution of the Cauchy problem (1.1) and of two-point boundary value problems related to the system arising in (1.1). Stability conditions for the solutions of the system of (1.1) are given. Because of developed techniques this paper can be regarded as a continuation of the sequence [3, 4, 5, 6].

scholarly journals Hyers-Ulam Stability of the First-Order Matrix Differential Equations

2015 ◽  
Vol 2015 ◽  
pp. 1-7
Author(s):  
Soon-Mo Jung
Keyword(s):  
Differential Equations ◽  
Variable Coefficients ◽  
Linear Differential Equations ◽  
Ulam Stability ◽  
Order Linear ◽  
First Order ◽  
Matrix Differential Equations ◽  
Homogeneous Matrix ◽  
Linear Differential ◽  
Matrix Differential

We prove the generalized Hyers-Ulam stability of the first-order linear homogeneous matrix differential equationsy→'(t)=A(t)y→(t). Moreover, we apply this result to prove the generalized Hyers-Ulam stability of thenth order linear differential equations with variable coefficients.

scholarly journals Some Results On The Asymptotic Behavior Of Linear Systems

1955 ◽  
Vol 7 ◽  
pp. 531-538 ◽  
Author(s):  
M. Marcus
Keyword(s):  
Differential Equation ◽  
Asymptotic Behavior ◽  
Matrix Differential Equation ◽  
Complex Matrix ◽  
Continuous Complex ◽  
Matrix Differential ◽  
Image Position ◽  
Vector Matrix ◽  
Complex Valued ◽  
Square Complex

1. Introduction. We consider first in §2 the asymptotic behavior as t → ∞ of the solutions of the vector-matrix differential equation(1.1) ,where A is a constant n-square complex matrix, B{t) a continuous complex valued n-square matrix defined on [0, ∞ ), and x a complex n-vector.

scholarly journals Double Tracking Control for the Directed Complex Dynamic Network via the Observer of Outgoing Links

2021 ◽  
Author(s):  
Peitao Gao ◽  
Yinhe Wang ◽  
Lizhi Liu ◽  
Lili Zhang ◽  
Xiao Tang
Keyword(s):  
Dynamic Model ◽  
Tracking Control ◽  
Dynamic Network ◽  
Complex Dynamic ◽  
System Perspective ◽  
Tracking Controllers ◽  
The Matrix ◽  
Matrix Differential Equations ◽  
Matrix Differential ◽  
Vector Differential Equation

Abstract From the large system perspective, the directed complex dynamic network is considered as being composed of the nodes subsystem (NS) and the links subsystem (LS), which are coupled with together. Different from the previous studies which propose the dynamic model of LS with the matrix differential equations, this paper describes the dynamic behavior of LS with the outgoing links vector at every node, by which the dynamic model of LS can be represented as the vector differential equation to form the outgoing links subsystem (OLS). Since the vectors possess the flexible mathematical operational properties than matrices, this paper proposes the more convenient mathematic method to investigate the double tracking control problems of NS and OLS. Under the state of OLS can be unavailable, the asymptotical state observer of OLS is designed in this paper, by which the tracking controllers of NS and OLS are synthesized to ensure achieving the double tracking goals. Finally, the example simulations for supporting the theoretical results are also provided.

Sign in / Sign up

Export Citation Format

Share Document