Input-to-State Stability for PDEs by Iasson Karafyllis & Miroslav Krstic

Author:Iasson Karafyllis & Miroslav Krstic
Language: eng
Format: epub
Publisher: Springer International Publishing, Cham

(5.3.27)

where (5.3.25) implies for all and that

(5.3.28)

Using (5.3.27), (5.3.28) and the fact that , we get for all :

(5.3.29)

(2)Estimate for .

Using integration by parts, we obtain from (5.3.26) for all and :

(5.3.30)

where , . Since , are orthonormal bases of , it follows that Parseval’s identity holds, i.e.,

(5.3.31)

(5.3.32)

It follows from (5.3.30) that the following inequality holds for all , and :

(5.3.33)

Using the Cauchy–Schwarz inequality and (5.3.33) we obtain for all , and :

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