Lyapunov equation
Usually, the matrix equation
![]() | (a1) |
where the star denotes transposition for matrices with real entries and transposition and complex conjugation for matrices with complex entries; is symmetric (or Hermitian in the complex case; cf. Hermitian matrix; Symmetric matrix). In fact, this is a special case of the matrix Sylvester equation
![]() | (a2) |
The main result concerning the Sylvester equation is the following: If and
have no common eigenvalues, then the Sylvester equation has a unique solution for any
.
When and there are no eigenvalues
of
such that
whatever
and
are (in the numbering of eigenvalues of
), then (a1) has a unique Hermitian solution for any
. Moreover if
is a Hurwitz matrix (i.e. having all its eigenvalues in the left half-plane, thus having strictly negative real parts), then this unique solution is
![]() | (a3) |
and if , then
. From this one may deduce that if
and
satisfy
, than a necessary and sufficient condition for
to be a Hurwitz matrix is that
. In fact, this last property justifies the assignment of Lyapunov's name to (a1); in Lyapunov's famous monograph [a1], Chap. 20, Thm. 2, one finds the following result: Consider the partial differential equation
![]() | (a4) |
If has eigenvalues with strictly negative real parts and
is a form of definite sign and even degree, then the solution,
, of this equation will be a form of the same degree that is sign definite (with sign opposite to that of
. Now, if
with
, then
, with
, is a solution of (a1). In fact,
is a Lyapunov function for the system
![]() | (a5) |
These facts and results have a straightforward extension to the discrete-time case: for the system
![]() | (a6) |
one may consider the quadratic Lyapunov function as above (i.e. ) and obtain that
has to be a solution of the discrete-time Lyapunov equation
![]() | (a7) |
whose solution has the form
![]() | (a8) |
provided the eigenvalues of are inside the unit disc.
The equation may be defined for the time-varying case also. For the system
![]() | (a9) |
one may consider the quadratic Lyapunov function and obtain that
has to be the unique solution, bounded on the whole real axis, of the matrix differential equation
![]() | (a10) |
This solution is
![]() | (a11) |
being the matrix solution of
,
. The solution is well defined if
defines an exponentially stable evolution (
,
). It is worth mentioning that if
and
are periodic or almost periodic, then
defined by (a11) is periodic or almost periodic, respectively. Extensions of this result to a discrete-time or infinite dimensional (operator) case are widely known. Actually, the Lyapunov equation has many applications in stability and control theory; efficient numerical algorithms for solving it are available.
References
[a1] | A.M. Lyapunov, "General problem of stability of motion" , USSR Acad. Publ. House (1950) (In Russian) |
[a2] | R.E. Bellman, "Introduction to matrix-analysis" , McGraw-Hill (1960) |
[a3] | A. Halanay, "Differential equations: stability, oscillations time lags" , Acad. Press (1966) |
[a4] | A. Halanay, D. Wexler, "Qualitative theory of pulse systems" , Nauka (1971) (In Russian) |
[a5] | A. Halanay, V. Räsvan, "Applications of Lyapunov methods in stability" , Kluwer Acad. Publ. (1993) |
Lyapunov equation. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Lyapunov_equation&oldid=50042