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50 Chapter 2. Global aspects 2.5 Global existence results
In this section we provide general global existence results for solutions of Riccati differential equations. These results will yield sufficient conditions such that the solution of a given Riccati differential equation with prescribed initial or terminal data exists in a given fixed time interval. In particular this interval may be unbounded.
Since some of the methods also apply to more general differential equa- tions, we shall also consider perturbed or generalized matrix Riccati differential equations. As already mentioned in Section 2.4, these sufficient existence con- ditions also imply that in the representation formula (2.14) the denominator is free of zeros in the considered time interval or, equivalently, det Q(t) ̸= 0 in (2.9).
In this section we shall make use of the following technical result:
Theorem 2.5.1. If K : I → Cr×r is piecewise continuously differentiable and Hermitian on an interval I ⊂ R, then its minimal eigenvalue λ is an absolutely continuous (in fact a Lipschitzian) function of t ∈ I. At any point where it is differentiable (therefore almost everywhere on the interval) it holds that
λ˙(t) = x∗(t)K˙ (t)x(t), (2.45) where x(t) is a unit eigenvector of K(t) corresponding to this minimal eigen-
value. The same also holds true for the maximal eigenvalue of K.
Proof. Since the proof for the maximal eigenvalue follows by a completely analo- gous argumentation, we restrict ourselves to the case of the minimal eigenvalue. Without loss of generality we may reduce our consideration to a (closed finite) interval on which K(·) is continuously differentiable. Let λ(t) be the minimal eigenvalue of K(t) and x(t) be a unit eigenvector of K(t) corresponding to this minimal eigenvalue. Let us introduce a function Λ(s,t) = x∗(t)K(s)x(t). Ev- idently for each t the function s → Λ(s,t) is continuously differentiable and λ(t) = Λ(t, t). As the minimal eigenvalue also is defined by
λ(s) = min x∗K(s)x, |x|=1
we infer that Λ(s, t) ≥ λ(s) = Λ(s, s) for all s, t. Then we may conclude for any t′, t′′:
Λ(t′′, t′′) − Λ(t′, t′′) = λ(t′′) − Λ(t′, t′′)
≤ λ(t′′) − λ(t′) ≤ Λ(t′′, t′) − λ(t′) = Λ(t′′, t′) − Λ(t′, t′),
deriving from it
t′′ ∂Λ ′′ ′′ ′
t′′ ∂Λ ′
(σ,t)dσ.
t′ ∂s
(σ,t )dσ≤λ(t )−λ(t)≤
t′ ∂s
(2.46)