 
Summary: NUMERICAL LINEAR ALGEBRA WITH APPLICATIONS
Numer. Linear Algebra Appl. 2009; 00:128 Prepared using nlaauth.cls [Version: 2002/09/18 v1.02]
A STABILIZATION ALGORITHM BASED ON ALGEBRAIC
BERNOULLI EQUATION
L. Amodei1, and J.M. Buchot1
1Institut de Math´ematiques de Toulouse  Universit´e Paul Sabatier  118 route de Narbonne  31062
Toulouse Cedex 9  France
SUMMARY
In this paper we give an explicit expression in a factorized form of the unique stabilizing solution of
the Algebraic Bernoulli Equation (ABE). It is deduced from the general expression of the solution of
the Algebraic Riccati Equation (ARE) introduced in [1]. More precisely, we show that this stabilizing
solution is of the form X = Z R1
Z
where Im(Z) is the invariant subspace associated to the unstable
eigenvalues of AT
and R the positive definite solution of a Lyapunov equation of dimension equal to
rank(Z). We show that the solution of the ARE converges to the stabilizing solution of the ABE when
the observation operator decreases to zero. This property allows to compare the impact of different
feedback laws on the spectrum of the closedloop matrix. The solution of the ABE can be used to
stabilize linear and nonlinear largescale systems. In particular, we show that the linear feedback
