Matrix differential equation
An equation in which the unknown is a matrix of functions appearing in the equation together with its derivative.
Consider a linear matrix differential equation of the form
![]() | (1) |
where is an
-dimensional matrix function with locally Lebesgue-integrable entries, and let
be an absolutely-continuous solution of equation (1) satisfying the condition
, where
is the identity matrix. Then the vector function
,
, is a solution of the linear system
![]() | (2) |
satisfying the condition . Conversely, if
and
is a solution of the system (2) satisfying the condition
,
, then the matrix
with as columns the solutions
is a solution of the matrix differential equation (1). If, in addition, the vectors
are linearly independent, then
for all
.
Equation (1) is a particular case of the following matrix differential equation (arising in the theory of stability)
![]() | (3) |
The solution of (3) with initial condition is given by the formula
![]() |
where is the solution of (1) with the condition
, and
is the solution of the matrix differential equation
with the condition
.
In various applied problems (theories of stabilization, optimal control, filtration of control system, and others) an important role is played by the so-called matrix Riccati differential equation
![]() |
Thus, if the matrix Riccati equation
![]() |
where stands for transposition, has for
a bounded solution
on the line
, and if for all
, all
, and some
, the inequality
holds, then every solution of the controllable system
![]() |
closed by the feedback law , satisfies the inequality
![]() |
where is the Euclidean norm and
does not depend on
.
References
[1] | I.A. Lappo-Danilevsky, "Mémoire sur la théorie des systèmes des équations différentielles linéaires" , Chelsea, reprint (1953) |
[2] | Yu.L. Daletskii, M.G. Krein, "Stability of solutions of differential equations in Banach space" , Amer. Math. Soc. (1974) (Translated from Russian) |
[3] | F.V. Atkinson, "Discrete and continuous boundary problems" , Acad. Press (1964) |
[4] | W.T. Reid, "Riccati differential equations" , Acad. Press (1972) |
[5] | M.Kh. Zakhar-Itkin, "The matrix Riccati differential equation and the semi-group of linear fractional transformations" Russian Math. Surveys , 28 : 3 (1973) pp. 89–131 Uspekhi Mat. Nauk , 28 : 3 (1973) pp. 83–120 |
Comments
References
[a1] | J.K. Hale, "Ordinary differential equations" , Wiley (1969) |
Matrix differential equation. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Matrix_differential_equation&oldid=40765