Page 21 - Textos de Matemática Vol. 36

Chapter 2
Global aspects of Riccati differential equations
2.1 Solutions of ARE and graph subspaces
We assume in the major part of this chapter that all coefficients of RDE or HRDE are constant and we write in this case M11, M12, ... instead of M11(t), M12(t), .... Throughout this section we consider the corresponding algebraic Riccati equations
and
0=M21 +M22W −WM11 −WM12W (ARE) 0 = −A∗X − XA − Q + XSX. (HARE)
In this chapter we are in particular interested in the structure of the set of all solutions of these algebraic Riccati equations; notice that each of these solutions is an equilibrium of the corresponding differential equation.
It is worthwhile to mention that not every first order quadratic matrix differential equation is (at least in our notation) a matrix Riccati differential equation; the main reason for restricting here to the class RDE is a result of Radon, [Rado27], [Rado28], which implies that matrix Riccati equations have very nice, characteristic properties. Radon found out (in the symmetric case) that RDE and the following linear system L of differential equations are related via a fractional transformation:
where M =
M11 M12 Y1
, Y1(t) ∈ C
n×n
and W0, Y2(t) ∈ C
M
M , Y = Y 2122 2
.
˙ In
,
(L)
m×n
Y =M(t)Y, Y(t0)= W
0
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