Periodic Feedback Stabilization for Linear Periodic Evolution Equations by Gengsheng Wang & Yashan Xu

Author:Gengsheng Wang & Yashan Xu
Language: eng
Format: epub
Publisher: Springer International Publishing, Cham

Proof

Let . Then there is and so that . Define , . It is clear that and . From these, we find that

This ends the proof.

Remark 3.1

By Theorem 1.​4, we see that Eq. (3.1) is linear periodic feedback stabilizable if and only if there are positive definite matrices Q and R so that the corresponding LQ problem satisfies the FCC for any . On the other hand, from Lemma 3.1, we find that for any positive definite matrices Q and R, the corresponding LQ problem satisfies the FCC for any x in . Thus, Eq. (3.1) is linear periodic feedback stabilizable if and only if there are positive definite matrices Q and R so that the corresponding LQ problem satisfies the FCC for any . The key now is to give a condition ensuring the existence of positive definite matrices Q and R so that the corresponding LQ problem satisfies the FCC for any .

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