Page 26 - Textos de Matemática Vol. 36

16 Chapter 2. Global aspects Q n×n
ii) If P is a real solution of the system L such that Q(t) ∈ R is regular for t ∈ [t0,tf] ⊂ R, then
W : [t0,tf] → Rm×n, t → P(t)Q−1(t) = W(t) (2.9) is a real solution of RDE.
iii) In case of a complex parameter t and M11 ∈ Cn×n, M12 ∈ Cn×m, M21 ∈ Cm×n, M22 ∈ Cm×m, and also W(t) ∈ Cm×n, the assertions i), ii) remain valid if we replace therein the interval [t0,tf] by an arbitrary domain G⊂C with t0 ∈G.
Since Radon’s Lemma provides a general relation between linear and non- linear differential equations of type RDE, we state
Remark 2.2.2. If the coefficients M11(t) ∈ Rn×n, M12(t) ∈ Rn×m, M21(t) ∈ Rm×n, M22(t) ∈ Rm×m, are locally integrable piecewise continuous matrix–valued functions defined on the interval [t0,tf] ⊂ R, then i) and ii) of Theorem 2.2.1 remain valid, in particular (2.9) yields a representation of the solutions of the matrix Riccati differential equation
W˙ =M21(t)+M22(t)W−WM11(t)−WM12(t)W in the interval [t0, tf ].
This shows that any matrix Riccati differential equation is locally equiv- alent to the linear system L. On the other hand it is clear from (2.9) that the solutions W of RDE may show the finite escape time phenomenon, i.e., the so- lutions may blow up on a finite interval. Therefore the most important problem in the theory of Riccati differential equations is to find sufficient conditions en- suring that the solutions of RDE do not blow up on a given interval. We will discuss this topic further in Section 2.5.
Proof. We prove Remark 2.2.2 since for a real time parameter t ∈ [t0,tf] the assertions i) and ii) in Theorem 2.2.1 follow from it.
Let W be a solution of the matrix Riccati differential equation and let Q be the solution of the linear initial value problem as defined in i). Notice that under our assumptions on the coefficient matrices Mij(t), 1 ≤ i,j ≤ 2, there always exists a unique solution of this initial value problem. Then
d Q Q˙ M11Q+M12WQ Q dt P = W˙Q+WQ˙ = M21Q+M22WQ =M P ,
hence, L is fulfilled.