Representation and Control of Infinite Dimensional Systems, by Alain Bensoussan, Giuseppe da Prato, Michel C. Delfour,

By Alain Bensoussan, Giuseppe da Prato, Michel C. Delfour, Sanjoy K. Mitter

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Additional resources for Representation and Control of Infinite Dimensional Systems, 2nd Edition (Systems & Control: Foundations & Applications)

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AB .. An−1 B]) = n. No satisfactory result similar to the pole-assignment theorem is known in infinite dimensions. e. θ ∈ [−h, 0), φ1 ∈ L2 (−h, 0; Rn ). 1. If the system is controllable, then there exists a feedback control u(t) = Kx(t) such that the closed loop system dx (t) = (A + BK)x(t) dt is asymptotically stable. 6 Stabilizability and detectability The structure theorem of linear systems and the pole-assignment theorem motivate the introduction of the concepts of stabilizability and detectability.

24) can be majorized by 28 I-1 Control of Linear Differential Systems ∞ t 0 M e−α(t−s) K |Cx(s)| ds 2 dt. 0 Now introduce the function f (s) = M K e−αs, s ≥ 0, 0, otherwise and |Cx(s)|, 0, g(s) = s ≥ 0, otherwise. Now f ∈ L1 (−∞, ∞; R) and g ∈ L2 (−∞, ∞; R). Hence by Young’s inequality f ∗g L2 ≤ f L1 . g L2 , where (f ∗ g)(t) = ∞ −∞ t f (t − s)g(s) ds = M K e−α(t−s)|Hx(s)| ds. 0 This proves the theorem. 4. This theorem and its proof generalize to certain infinite dimensional Hilbert space situations and have implications in the study of the algebraic Riccati equation.

N }. Let xi , i = 1, . . , n, be the corresponding eigenvectors. Hence xi = (λi I − A)−1 BKxi , Now (λI − A)−1 = i = 1, 2, . . , n. n ρj (λ)Aj−1 , j=1 where ρj (λ) are rational functions defined on the complement C\σ(A) of the spectrum σ(A) of A. Hence 2 Controllability, observability, stabilizability, and detectability n xi = 23 . . AB .. An−1 B]). j=1 . . AB .. An−1 B]) = n. No satisfactory result similar to the pole-assignment theorem is known in infinite dimensions. e. θ ∈ [−h, 0), φ1 ∈ L2 (−h, 0; Rn ).

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