By Gengsheng Wang, Yashan Xu
This booklet introduces a couple of fresh advances concerning periodic suggestions stabilization for linear and time periodic evolution equations. First, it provides chosen connections among linear quadratic optimum keep an eye on idea and suggestions stabilization thought 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 one can layout periodic suggestions legislation. finally, the publication introduces readers to key tools for designing the keep an eye on machines. Given its assurance and scope, it bargains a precious advisor for graduate scholars and researchers within the parts of keep watch over conception and utilized mathematics.
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Additional info for Periodic Feedback Stabilization for Linear Periodic Evolution Equations
Example text
Jnˆ lˆnˆ ⎞ ⎟ ⎟ ⎟ a d jk × d jk matrix, ⎟ .. 1⎠ λ¯ j .. lˆ j where j = 1, . . , n, ˆ k = 1, . . , lˆj , and for each j, {d jk }k=1 is decreasing. It is clear lˆ j that k=1 d jk = l j for each j = 1, . . , n, ˆ and the basis β as nˆ j=1 lˆj k=1 d jk = n 0 . We rewrite , ξnˆ lˆnˆ 1 , . . , ξnˆ lˆnˆ d ξ111 , . . , ξ11d11 , ξ1lˆ1 1 , . . , ξ1lˆ1 d ˆ , . . , ξn11 ˆ , . . , ξn1d ˆ n1 ˆ β nˆ lˆnˆ 1l1 . 74), one can easily check that for each j ∈ {1, . . , n} ˆ and k ∈ {1, . . , lˆj }, ξ jk(r −q) 0 q λ¯ j I − Q ξ jkr = when r > q, when r ≤ q.
111), we find that Since Q kT q kT y(s; 0, h) 2 ds ≤ 0 yk,h (s), Q yk,h (s) ds ≤ h, Υ¯ (0)h for all k ∈ N and h ∈ H. 0 This leads to ∞ y(s; 0, h) 2 ds ≤ 1 Υ¯ (0) q for each h ∈ H. 112) Let Φ K be the evolution generated by A(·) + B(·) K¯ (·). Then sup Φ K (T, r ) ≤ 0 h 2 r ∈[0,T ] + C1 for some C1 > 0. Fix an h ∈ H . We have that for each s ∈ R , Φ K ([s/T ]T + T, 0) h ≤ Φ K (T, s − [s/T ]T ) Φ K (s, 0) h ≤ C1 Φ K (s, 0) h . Hence, we find that Φ K (s, 0) h ≥ Φ K ([s/T ]T + T, 0) h /C1 for each s ∈ R+ .
Taking the infimum on the both sides of the above equation with respect to u(·) ∈ L 2 (R+ ; U ) leads to W ∞ (t, h) = W ∞ (0, h). So the value function W ∞ (t, h) is independent of t. This completes the proof. 3, we see that linear time-periodic functions K (·) will not aid in the linear stabilization of Eq. 1) when both D(·) and B(·) are time invariant. On the other hand, when Eq. 1) is T -periodically time varying, linear timeperiodic functions K (·) do aid in the linear stabilization of Eq. 1).