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Backstepping Control
Published in Bogdan M. Wilamowski, J. David Irwin, Control and Mechatronics, 2018
By using the Lasalle’s theorem, this Lyapunov function (20.40) guarantees the global uniform boundedness of z1, z2, θ^1, θ^2, and z1, z2 → 0 as t → ∞. It follows that asymptotic tracking is achieved, such that limt→∞(x1 – xr) = 0. Since z1 and xr are bounded, x1 is also bounded from x1 = z1 = xr. The boundedness of x2 follows from boundedness of x˙r and a1 in (20.26) and the fact that x2=z2+α1+x˙r. Combining this with (20.35), we conclude that the control u(t) is also bounded.
Steinhaus Type Theorems over Valued Fields: A Survey
Published in Michael Ruzhansky, Hemen Dutta, Advanced Topics in Mathematical Analysis, 2019
Remark 16.2.3. In the context of Theorem 16.2.2, it is worth noting that the absence of an analogue in non-archimedean analysis for the signum function in classical analysis possibly made Monna Monna (1963) use functional analytic tools like the analogue of Uniform Boundedness Principle to prove Theorem 16.2.2. However, Natarajan Natarajan (1991) proved Theorem 16.2.2 without using functional analytic tools.
Rate Control Design
Published in Christos N. Houmkozlis, George A. Rovithakis, End-to-End Adaptive Congestion Control in TCP/IP Networks, 2017
Christos N. Houmkozlis, George A. Rovithakis
In this section, we shall present a systematic tool, based upon a Lyapunov function derivative estimation approach, for the design of controllers capable to guarantee a uniform ultimate boundedness property for the tracking error e = RTT — RTTd, as well as the uniform boundedness of all other signals in the closed-loop.
Subdifferentials and derivatives with respect to a set and applications to optimization
Published in Applicable Analysis, 2019
Vo Duc Thinh, Thai Doan Chuong
for all and where can be chosen such that Let Taking with and , we assert that On the one hand, since as , it follows that is bounded in due to Banach–Steinhaus uniform boundedness principle. Hence,