It creates also 4 compound terms of the 2nd (previous) hidden layer, using reverse outputs and inputs of 2 bound blocks in respect to 4 derivative variables (12).
Then the nonlinear dynamical system (3.1), (3.2) is ultimately bounded with respect to uniformly in with ultimate bound where.
If, in addition, and is a class- function, then the nonlinear dynamical system (3.1) and (3.2) is globally ultimately bounded with respect to uniformly in with ultimate bound.
(ii) The discrete-time nonlinear dynamical system (3.1) and (3.2) is ultimately bounded with respect to uniformly in with ultimate bound if there exists such that, for every, there exists such that implies,.
Hence, it follows from Theorem 3.3 that the closed-loop system given by (4.17), (A.2), and (A.3) is globally ultimately bounded with respect to uniformly in with ultimate bound given by, where.
The discrete-time nonlinear dynamical system (3.1) and (3.2) is globally ultimately bounded with respect to uniformly in with ultimate bound if, for every, there exists such that implies,. .
If there does not exist such that for all, it follows using similar arguments as in the proof of Theorem 4.1 that the closed-loop system (5.4), (B.3), and (B.4) is globally ultimately bounded with respect to uniformly in with ultimate bound given by, where.
Martin boundaries and integral representations of positive functions which are harmonic in a bounded domainDwith respect to Brownian motion are well understood.
We claim that ({varphi(t_{n},u)}_{ninmathbb{N}}) is uniformly bounded in H with respect to (ninmathbb{N}).
Since (2.4) contradicts (2.5), we find that ({varphi (t_{n},u)}_{ninmathbb{N}}) is uniformly bounded in H with respect to (ninmathbb{N}) and the claim is proved.
