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Definite Integral
Published in John Srdjan Petrovic, Advanced Calculus, 2020
Since f is a bounded function, its range is a bounded set. Thus the numbers Mi = sup{f(x) : x ∈ [xi−1, xi]} and mi = inf{f(x) : x 2 [xi−1, xi]} are well defined. The sums L(f,P)=∑i=1nm1ΔxiandU(f,P)=∑i=1nm1Δxi, are the lower and the upper Darboux sums.
Variables, functions and mappings
Published in Alan Jeffrey, Mathematics, 2004
A function y = f(x) is said to be bounded on an interval if it is never larger than some value M and never smaller than some value m for all values of x in the interval. An example is shown in Fig. 2.9(f). The numbers M and m are called, respectively, upper and lower bounds for the function f(x) on the interval in question. It may of course happen that only one of these conditions is true, and if it never exceeds M then it is said to be bounded above, whereas if it is never less than m it is said to be bounded below. A bounded function is thus a function that is bounded both above and below. The bounds M and m need not be strict in the sense that the function ever actually attains them. Sometimes when the bounds are strict they are only attained at an end point of the domain of definition of the function.
Two adaptive nonmonotone trust-region algorithms for solving multiobjective optimization problems
Published in Optimization, 2023
Nasim Ghalavand, Esmaile Khorram, Vahid Morovati
For all , The following conditions are equivalent:The point x is not critical,,. is a continuous function and is a bounded function on any compact set.
Existence of asymptotically periodic solutions of partial functional differential equations with state-dependent delay
Published in Applicable Analysis, 2021
Filipe Andrade, Claudio Cuevas, Hernán R. Henríquez
It only remains to prove that . Let . It follows from (10) that Moreover, is a bounded function on , and combining with condition (F1) follows that is a bounded set. Also for . The first term in right hand side of the above decomposition since f is uniformly S-asymptotically ω-periodic on bounded sets. For the second term, using (F2), (R1) and (R2), we obtain Let . Combining these assertions, we infer that the function . Hence, it follows from Lemma 4.2 that , which completes the proof.
An asinh-type regulator for robot manipulators with global asymptotic stability
Published in Automatika, 2020
Fernando Reyes-Cortes, Basil M. Al-Hadithi
This section describes the main properties and qualities of the new asinh-type control scheme, which we denote by asinh. Our motivation to use this function as control structure is based on the following: This function is a vectorial map, ; being continuous in . It is not a bounded function. But it is a monotonically increasing function and located within the first and third quadrants; it is an odd function and symmetric about the origin. It is also approximately linear close to the origin. When takes large values, its behavior is similar to , as shown in Figure 1. As approaches positive (negative) infinity, approaches positive (negative) infinity, respectively. However, its first partial derivative (gradient , for ) is bounded. Therefore, its growth rate is slower than the PD control and exponential increasing of the hyperbolic functions (sinh and cosh functions). All these qualities represent key elements to design the control structure, as well as tuning the control gains and avoiding the saturation of the servo-amplifiers. The above features are used to drive the position error and to obtain energy dissipation from the derivative term.