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.
In Section 3, under suitable hypotheses, we prove that the problem (1.1) possesses at least three solutions when lies in exactly determined two open intervals, respectively; moreover, all these solutions are uniformly bounded with respect to belonging to one of the two open intervals.
The form t is called relatively form bounded with respect to (mathbf{t}_{A}) ((mathbf{t}_{A} -bounded) if (operA} -boundedm}(mathbf {t}_{A})subset operatorname{Dom}(mathbf{t})) and there are posifive coperatorname{ such that bigl|mathbf{t}[f]bigr|leqslant amathbf{t}_{A}[f]+b| f |_{mathfrak{H} }^{2},quaDominoperatorname{Dom}( mathbf{t}_{A}).
For example, we show thatSα, βis bounded with respect to a weightWif and only ifWbelongs toHSror |α−β|W≡0.
The state estimation error of the proposed high-gain approximate differentiator based sliding-mode observer is shown to be uniformly ultimately bounded with respect to a ball whose radius is a function of design parameters.
On the other hand, it is statistically bounded with respect to.
