Riccati matrix
equations in stability of linear difference equations
A.A. Kovalev, V.B. Kolmanovskii
Moscow State Institute of Electronics and Mathematics
109028, Bol. Trekhsvyatitelskii 3/12, Moscow, Russia.
Difference equations are used to describe many physical, biological and
other processes Stability is an important property of solutions of these
equations. The powerful tool for investigating stability of difference
equations is direct Lyapunov method which consists of constructing some
functionals defined on the solutions of the equations concerned and satisfying
the conditions of corresponding theorems.
Consider the difference equation
, 
(1)
where
is a
constant matrix 
.
It is well known that the necessary
and sufficient condition for the asymptotic stability of (1) is the existence
of positive definite symmetric matrix 
which
is a unique solution of the matrix equation
, (2)
for any positive definite symmetric matrix 
.
Consider now the simplest difference equation with
delay 
, 
(3)
For stability investigation of (3) we can apply the scheme (1), (2). In
fact we can introduce new phase vector 
and
rewrite (3) in the form (1). After this stability condition in the form (2) can
be used.
Two obstacles arise here:
1)
The dimension of 
tends
to 
as 
(it
means that the dimension of the matrix 
tends to 
).
2)
If we try to investigate robust stability of
(3) with respect to 
(it
means independently on the values of 
) then we have to check infinite number
conditions of the form (2) (i.e. for each value of 
).
Other way to investigate stability of (1) is connected with the
investigation of the location of the roots of the characteristic equation
corresponding to (1). But the attempt to generalize this approach for (3) is
connected with the same obstacles 1) and 2) mentioned above.
Here we propose for stability investigation of difference equations with
delay to use approach connected with the using of Riccati matrix equations for
the matrix 
. This approach, used in for investigation the
stability of some discrete Volterra equations and also in for linear stochastic
differential equations with delay, allows us to keep the dimension of these
matrix Riccati equations the same (
) for all values of delay 
and,
correspondingly, to obtain robust stability conditions with respect to 
. Besides for some concrete classes of
equations the stability conditions formulated directly in terms of
characteristics of equations concerned. Moreover, various Riccati equations can
be obtained for the same original equation with delay using various methods of
the transformation of this equation and various auxiliary Lyapunov functions. Note
that all these matrix Riccati equations coincide with (2) if delay would be absent.
Also the greater number of Riccati equations we have the greater part of
stability domain could be obtained.
The basic tool to obtain Riccati equations is connected with some
Lyapunov like theorem. The necessary procedures are described.