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2.5. Global existence results 53 Letusputnowη1 =x∈Rn, η2 =Wx∈Rm;thenweobtain
˙ −λ(M +MT) MTC−λM η λ=(ηT,ηT) 11 11 21 12 1 .
1 2 CM −λMT CM +MTC η 21 12 22 22 2
Assume that there is a t1 < t0 such that λ(t1) ≥ a. With W0 := W(t0,W0) and ||W0||C < √a, i.e., λ(t0) < a, from continuity of λ we infer the existence of t∗ ∈[t1,t0]suchthatλ(t∗)=aandsuchthatforallδ>0itholdsthatλ˙(t)<0 in some subinterval of (t∗, t∗ + δ).
Let δ be chosen sufficiently small; then λ(t) is so close to a such that from (2.47) we infer again by a continuity argument that
−λ(M +MT) MTC−λM
11 11 21 12 (t) > 0
CM −λMT CM +MTC 21 12 22 22
for t ∈ [t∗, t∗ +δ]. This yields a contradiction and proves λ(t) < a or equivalently ||W(t,W0)||C < √a for all t ≤ t0.
From the proof it can also be seen that the set ||W||C > √a is positive invariant. It should be pointed out that this method also can be used in the in- finite dimensional case, i.e., in the case of operator Riccati differential equations (for details see [Kuip94]).
As a first application we use the general results from Theorem 2.5.2 to ob- tain global existence results for solutions of standard Riccati differential equa- tions HRDE (see Chapter 4). They appear in standard linear quadratic optimal control problems and their generalizations in H∞-problems. A detailed analy- sis of this important class of Riccati differential equations will be presented in Chapter 4.
Corollary 2.5.3. Let us consider the HRDE
P˙ =−ATP−PA−Q+PSP, (2.48)
where A,Q,S,P(t) ∈ Rn×n and Q = QT, S = ST. The associated algebraic Riccati equation HARE is
0=−ATP −PA−Q+PSP.
Let furthermore the reverse dichotomic solution Prd of (2.49) exist. If
(2.49)
(2.50)
turns out to be positive definite for some a > 0, then the ball ||P − Prd|| < √a remains negative invariant for solutions P of (2.48).
−a(A+AT −(SPrd +PrdS)) aS aS −(A + AT − (SPrd + PrdS))