Page 44 - Textos de Matemática Vol. 36

34 Chapter 2. Global aspects of measure zero. Consequently, the flow of the Riccati equation can be regarded
loosely as a linear flow composed with the projection
Π:C
(m+n)×n n m+n Q Q Q −→G(C ), P −→Π P =Im P .
The reason that we say loosely is that the identification between W(t,W0,t0) and
S(t, ψ(W0), t0) is valid only as long as S(t, ψ(W0), t0) ∈ Gn0 (Cm+n).
This suggests a two–step strategy to characterize the phase portrait of the Riccati differential equation:
(1) Determine the phase portrait of the flow S(t, S0, t0) on Gn(Cm+n).
(2) Identify which of the important dynamic phenomena (e.g., equilibrium points, periodic orbits, invariant tori etc. ) are actually contained in the subset
Gn0 (Cm+n) and hence represent dynamic phenomena for the Riccati differential equation on the space of matrices.
The first step has been carried out in case of constant coefficients in [Shay86] under the condition of simple eigenvalues and under the additional restriction that eigenvalues having the same real part are conjugate complex to each other. The phase portrait of the so–called extended Riccati differen- tial equation ERD was studied there, see also [CaWi81], [Shay86], [HeMa82], [HeMa83] and the literature cited therein.
The second step was also studied in [Shay86] but was restricted to the case of symmetric Riccati differential equations appearing in control theory under system theoretic assumptions (e.g., controllability). Here we will build on our representation formula (2.14) in order to obtain information on the phase por- trait. Clearly the trajectory S(t, ψ(WC ), 0) = S(t, ψ(WC )) belongs to Gn0 (Cm+n) as long as, in (2.14),

detQ(t) = 1≤k1<···<kn≤m+n
e
(λk1+···+λkn)t
. |v˜k1,...,v˜kn| .
c kn

̸= 0.
Therefore in the forthcoming Section 2.5 we will present conditions on the co- efficients and initial data of RDE such that det Q(t) does not vanish.
But first, we utilize the representation formula (2.14) for the representation of the solutions of the algebraic Riccati equation ARE. We have already seen by our examples that if we want full information about all solutions of ARE we must deal with complex solutions and a complex parameter t.
We introduce the following notation to make (2.14) more practical: a) For
λ1,...,λm+n ∈ C and a = (a1,...,am+n) ∈ {0,1}m+n
c k1