On linear matrix differential equations

Abstract

We use elementary methods and operator identities to solve linear matrix differential equations and we obtain explicit formulas for the exponential of a matrix. We also give explicit constructions of solutions of scalar homogeneous equations with certain initial values, called dynamic solutions, that play an important role in the solution of homogeneous and non-homogeneous matrix differential equations. We show that the same methods can be used to solve linear matrix difference equations.

Solution to dynamic matrix linear systems by operator exponential functions

“…In this section, we give the state response of the matrix linear system (7), where X (t) ∈ R m×n is the system state, U (t) ∈ R r×p is the system input, A is the system operator given in (5), and B is the input operator given in (6).…”

“…In [5], the solution of Lyapunov differential matrix equations was given in terms of matrix exponential functions. In [6], elementary methods and operator identities were used to solve the simple linear matrix differential equations in the form ofẊ (t) = AX (t) + U (t). In [7], the Sylvester matrix control systems were investigated.…”

Variable structure interacting multiple model for highly aperiodic sensors with poor measurement precision

“…where r i is the rate at which the transition occurs when in state i, and q ij is called an instantaneous transition rate describing the rate of transition from state i to state j. Given boundary conditions P{m i (0)|m i (0)} = 1 and P {m i (0)|m j (0)} = 0, the solution to Kolmogorov's backward Equation ( 14) is [25]:…”

The matrix of matrices exponential and application

“…The asymptotic properties of matrix exponential functions extend information on the long-time conduct of solutions of ODEs. (see [18,9,21,1,26]). The concept of the MMs was introduced by Kishka et al (cf.…”