Page 67 - Textos de Matemática Vol. 36

2.5. Global existence results 57 or after a symmetrization as
˙∗∗∗∗ ∗Qx V=(xQ,xP)(Θ+Θ) Px , (2.58)
where Θ was defined in (2.53). From Theorem 2.2.1, (2.56), (2.58), (2.53) and (2.54) we conclude that V˙ (t) is piecewise continuous and V˙ (t) ≤ 0 in (−∞, t0 ]. Hence, V (t) is monotonically decreasing in (−∞, t0 ].
Assume now that C + DW0 + W0∗D∗ > 0 holds; then from Theorem 2.2.1 and formulas (2.56), (2.58) and (2.55) we infer
V(t0)=x∗(C+DW0 +W0∗D∗)x>0,
which together with the monotonicity of V (t) implies that V (t) > 0 for all t ≤ t0. Hence, Q(t) must be regular in (−∞, t0], which implies the existence of W (t, W0) in (−∞, t0].
The condition (2.55) is rather restrictive, even in particular applications, as for example in the standard control theoretic case. Therefore we suggest a refined approach.
Corollary 2.5.7. If with the notation of Theorem 2.5.6 condition (2.54) holds and if instead of (2.55) also it holds that
and
 rank
C + D W 0 + W 0∗ D ∗ ∗ ∗

I =n,
C + DW0 + W0∗D∗ ≥ 0
(2.59)
(2.60) then the solution W (t, W0) of RDE with W (t0, W0) = W0 exists for all t ≤ t0.
Proof. Integrating equation (2.58) from t ≤ t0 to t and observing that V (t0) = x∗(C + DW0 + W0∗D∗)x, we obtain
I
W0 (Θ(t0) + Θ (t0))
W0
V(t)=V(t,x) = x∗(C+DW0+W0∗D∗)x
−x∗ ( t0Q∗ (s) P ∗ (s) (Θ(s) + Θ∗ (s)) Q(s) ds)x. (2.61)
t P(s) From the assumptions (2.54) and (2.59) it follows that
and
x ∗ ( C + D W 0 + W 0∗ D ∗ ) x ≥ 0
t0 Q(s)
−x∗(
Q∗(s), P ∗(s) (Θ(s) + Θ∗(s)) P (s) ds)x ≥ 0.
t