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

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.

Show description

Read Online or Download Periodic Feedback Stabilization for Linear Periodic Evolution Equations PDF

Similar linear books

Lie Groups and Algebras with Applications to Physics, Geometry, and Mechanics

This ebook is meant as an introductory textual content with regards to Lie teams and algebras and their position in quite a few fields of arithmetic and physics. it really is written by means of and for researchers who're basically analysts or physicists, no longer algebraists or geometers. now not that we've got eschewed the algebraic and geo­ metric advancements.

Dimensional Analysis. Practical Guides in Chemical Engineering

Sensible publications in Chemical Engineering are a cluster of brief texts that every offers a centred introductory view on a unmarried topic. the entire library spans the most subject matters within the chemical strategy industries that engineering pros require a easy realizing of. they're 'pocket guides' that the pro engineer can simply hold with them or entry electronically whereas operating.

Linear algebra Problem Book

Can one examine linear algebra completely by way of fixing difficulties? Paul Halmos thinks so, and you'll too when you learn this e-book. The Linear Algebra challenge ebook is a perfect textual content for a direction in linear algebra. It takes the coed step-by-step from the elemental axioms of a box during the inspiration of vector areas, directly to complicated innovations comparable to internal product areas and normality.

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).

Download PDF sample

Rated 4.94 of 5 – based on 47 votes