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Concept
Matrix differential equation
Résumé
A differential equation is a mathematical equation for an unknown function of one or several variables that relates the values of the function itself and its derivatives of various orders. A matrix differential equation contains more than one function stacked into vector form with a matrix relating the functions to their derivatives.
For example, a first-order matrix ordinary differential equation is
where is an vector of functions of an underlying variable , is the vector of first derivatives of these functions, and is an matrix of coefficients.
In the case where is constant and has n linearly independent eigenvectors, this differential equation has the following general solution,
where λ1, λ2, ..., λn are the eigenvalues of A; u1, u2, ..., un are the respective eigenvectors of A; and c1, c2, ..., cn are constants.
More generally, if commutes with its integral then the Magnus expansion reduces to leading order, and the general solution to the differential equation is
where is an constant vector.
By use of the Cayley–Hamilton theorem and Vandermonde-type matrices, this formal matrix exponential solution may be reduced to a simple form. Below, this solution is displayed in terms of Putzer's algorithm.
The matrix equation
with n×1 parameter constant vector b is stable if and only if all eigenvalues of the constant matrix A have a negative real part.
The steady state x* to which it converges if stable is found by setting
thus yielding
assuming A is invertible.
Thus, the original equation can be written in the homogeneous form in terms of deviations from the steady state,
An equivalent way of expressing this is that x* is a particular solution to the inhomogeneous equation, while all solutions are in the form
with a solution to the homogeneous equation (b=0).
In the n = 2 case (with two state variables), the stability conditions that the two eigenvalues of the transition matrix A each have a negative real part are equivalent to the conditions that the trace of A be negative and its determinant be positive.
Source officielle
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