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The state–output system is said to be asymptotically output stable if ∞ |CeAt x|2 dt < ∞. ∀x ∈ Rn , 0 2 Controllability, observability, stabilizability, and detectability 27 Now it is easy to see that if (A, C) is an observable pair and A is asymptotically stable, then the equation A∗ P + P A = −C ∗ C has a solution and ∞ P = ∗ eA t C ∗ CeAt dt > 0 (positive deﬁnite), 0 the positive-deﬁniteness being a consequence of observability. Conversely if the state–output system is asymptotically output stable and if (A, C) is an observable pair, then A is asymptotically stable.

2. Let G(s) = C(sI − A)−1 B with A stable. Then G and only if there exists an X = X ∗ ≥ 0 that satisﬁes ∞ < 1 if XA + A∗ X + C ∗ C + XBB ∗ X = 0 with A + BB ∗ X stable. Furthermore, the state feedback u(t) = BB ∗ Xx(t) solves 1 ∞ sup (|y(t)|2 − |u(t)|2 ) dt. u 2 0 For a proof of the above lemma see J. C. Willems [1]. 1 is now presented. 3) with A + (LL∗ − BB ∗ )X stable. 3) yields X(A − BB ∗ X) + (A − BB ∗ X)∗ X + (C − DB ∗ X)∗ (C − DB ∗ X) + XLL∗X = 0. 4) Now assumptions (A2) to (A3) imply that the pair (A − BB ∗ X, C − DB ∗ X) is observable.

The following are true: (i) R(A) is dense in K ⇐⇒ N (A∗ ) = {0}. (ii) N (A) = {0} ⇐⇒ R(A∗ ) is dense in H. (iii) R(A) is dense in K ⇐⇒ AA∗ : K → K satisﬁes AA∗ > 0. (iv) N (A) = {0} ⇐⇒ A∗ A : H → H satisﬁes A∗ A > 0. (v) A ∈ L(H; K) is invertible ⇐⇒ R(A) = K, N (A) = {0} ⇐⇒ ∃c > 0, such that h ≤ c Ah , ∀h ∈ H, ∗ ⇐⇒ ∃c > 0, such that k ≤ c AA k , ∀k ∈ K. 1. Much of the above extends to operators A that are densely deﬁned and closed to spaces H and K, which are Banach spaces. For proofs of these facts, see, for example, M.