Periodic Feedback Stabilization for Linear Periodic by Gengsheng Wang, Yashan Xu

By Gengsheng Wang, Yashan Xu

This e-book introduces a couple of contemporary advances concerning periodic suggestions stabilization for linear and time periodic evolution equations. First, it offers chosen connections among linear quadratic optimum keep an eye on thought and suggestions stabilization idea for linear periodic evolution equations. Secondly, it identifies numerous standards for the periodic suggestions stabilization from the viewpoint of geometry, algebra and analyses respectively. subsequent, it describes numerous how you can layout periodic suggestions legislation. finally, the publication introduces readers to key tools for designing the regulate machines. Given its insurance and scope, it bargains a priceless advisor for graduate scholars and researchers within the parts of keep an eye on concept and utilized mathematics.

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Example text

Defined two functions u kt,h and z t,h by u kt,h (s) = u¯ kt,h (s), when s ∈ [t, k], 0, when s ∈ (k, ∞), 1 k z t,h (s) = k (s), when s ∈ [t, k], Q 2 y¯t,h 0, when s ∈ (k, ∞). Notice that ∞ t k k z t,h (s) 2 ds = t k k k y¯t,h (s), Q y¯t,h (s) ds ≤ Jt,h (u¯ kt,h ) = W k (t, h) and ∞ t u kt,h (s) 2 ds ≤ 1 δ k t u¯ kt,h (s), R u¯ kt,h (s) ds ≤ 1 k k 1 Jt,h (u¯ t,h ) = W k (t, h), δ δ where δ is a positive real number satisfying R ≥ δ I . 91), yields that {u kt,h , k ∈ N, k > t} and {z t,h , k ∈ N, k > t} are bounded in L 2 (t, ∞; U ) and L 2 (t, ∞; H ), respectively.

Clearly, it stands when k = 1. 11) AkZ0 = A1Z + PA1Z + · · · + P k0 −1 A1Z . 3), we have that Φ((k0 + 1)T, T ) = Φ(T, 0)k0 = P k0 . 11), indicates that AkZ0 +1 = (k0 +1)T Φ((k0 + 1)T, s)B(s)u(s)ds u(·) ∈ L 2 (R+ ; Z ) 0 = P A1Z + k0 T k0 Φ(k0 T, s)B(s)u(s + T )ds u(·) ∈ L 2 (R+ ; Z ) 0 = P k0 A1Z + AkZ0 = A1Z + PA1Z + · · · + P k0 A1Z . 8). 8). 23), we have that PP = PP. 8). 9). 8) that AˆnZ0 ⊆ AˆZ and AˆkZ ⊆ AˆnZ0 , when k ≤ n 0 . 6)), according to the j Hamilton-Cayley theorem, each P1 with j ≥ n 0 is a linear combination of I, P11 , (n −1) P12 , · · · , P1 0 .

1) is linear periodic feedback stablizable. 3 Relation Between Periodic Stabilization and LQ Problems 27 holds for some positive constants C and δ. 106) u h (s) = K (s)y K (s; 0, h), s ≥ 0. It is clear that y(s; 0, h, u h ) = y K (s; 0, h), s ≥ 0. 106), we find that C2 ∞ (u) ≤ Q + R K 2L ∞ (R+ ;L (H,U )) J0,h h 2. 2δ Hence, (L Q)∞ 0,h (corresponding to the pair (Q, R)) satisfies the FCC for any h ∈ H . 88) when Q 0 and R 0. We next show that (iii)⇒(i). Let Q 0 and R 0 so that the corresponding satisfies the FCC for all h ∈ H .

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