China 
Africa 
Explore chapters and articles related to this topic
Banach Spaces
Published in Eberhard Malkowsky, Vladimir Rakočević, Advanced Functional Analysis, 2019
Eberhard Malkowsky, Vladimir Rakočević
The third fundamental theorem is the uniform boundedness principle (Theorem 3.7.4 for pointwise bounded sequences of continuous linear functionals on a Fréchet space). It asserts that any set of pointwise bounded continuous linear maps from one Banach space to another is uniformly bounded on the unit ball. The uniform boundedness principle was proved by Hahn in 1922 for continuous linear functionals on a Banach space. Later it was proved by Hildebrandt for continuous linear maps between Banach spaces in 1923 and also proved in a more general version by Banach and Steinhaus in 1927. The uniform boundedness principle is also referred to as the Banach-Steinhaus theorem (Theorem 3.7.5 for pointwise convergent sequences of continuous linear functionals on a Fréchet).
Banach Spaces
Published in Hugo D. Junghenn, Principles of Analysis, 2018
The following theorem asserts that under suitable conditions a family of bounded linear transformations that is pointwise bounded is uniformly bounded on bounded sets. The proof depends on the Baire category theorem (0.3.12).
A Model-Constrained Tangent Slope Learning Approach for Dynamical Systems
Published in International Journal of Computational Fluid Dynamics, 2022
Assume that the second derivative of with respect to is uniformly bounded. Let and where . Then, the prediction error at time satisfies
Computing stabilising output-feedback gains for continuous-time linear time-varying systems through discrete-time periodic models
Published in International Journal of Control, 2021
C. M. Agulhari, P. L. D. Peres
Lastly, according to Agulhari et al. (2018), condition (4) is satisfied for a state-feedback control law if the pair is uniformly completely controllable, which is satisfied by hypothesis, and if the gain satisfies being a uniformly bounded function. Thus, the output-feedback control law guarantees the validity of (4) if the bound is satisfied. Note that If the discrete-time gains are bounded for all , and since is piecewise constant according to (7), then there exists a uniformly bounded function such that From Kalman (1960), the uniform complete observability of the pair , guaranteed by hypothesis, implies for an uniformly bounded function . Therefore, Thus, a necessary condition for the uniform complete observability is the existence of a uniformly bounded function satisfying As a consequence, inequality (17) is satisfied, and so is condition (4). Hence, according to Lemma 2.1, the control law (2) with the output-feedback gain given in (7) guarantees the uniform asymptotic stability of system (1).
Non-linear Control of Grid Tied Solar Photovoltaic System Considering Uncertainties
Published in IETE Journal of Research, 2023
Since, is differentiable with first-order derivative uniformly bounded. This satisfies is a constant and positive.