Page 12 - Textos de Matemática Vol. 36

2 Chapter 1. Introduction
where generally W is a complex m × n-matrix and where M11, M12, M21, M22 are constant complex matrices of dimensions n × n, n × m, m × n and m × m, respectively. As we have seen by the introductory example of a scalar equation, in order to obtain all information about stationary points (i.e., about solutions of ARE) it is necessary to deal with complex solutions even if the coefficients M11,M12,M21,M22 arereal.
In some discussions in this section we shall allow the coefficient matrices M11,M12,M21, M22 of RDE to be time dependent instead of being constant; in this case it does not make sense to speak of the associated algebraic Riccati equation ARE.
With this matrix Riccati equations we associate the blocked (m + n) × (m + n) matrix
M11 M12 M=MM.
21 22
Let us first briefly come back to the example from above (one-dimensional; complex):
where the matrix M becomes
M=10.
The equation has the solutions
w(z) = tan(z + c) = i(αeiz + βe−iz),
w′ =1+w2, 0 −1
αeiz − βe−iz
Notice that the solutions can be written as a quotient of solutions of linear dif-
αβ ̸= 0.
ferential equations.
For given initial data in t = 0 we obtain
w0=w(0)= α−β i(α+β)
and this determines (not uniquely) the parameters α,β.
To α = 0, or β = 0 correspond the constant solutions which are the stationary points w± = ±i of the dynamical system described by the differential equation. These stationary points clearly are solutions of the algebraic Riccati equation 1+w2.
If α ̸= 0, then:
1 − β e−2iz
lim α = −i.
Imz→−∞ i(1+ βe−2iz) α