The overall adaptive scheme guarantees the stability of the resulting closed-loop system in the sense that all the states and signals are uniformly bounded and arbitrary small attenuation level of the external disturbance on the tracking error can be achieved.
Moreover, the stability analysis of the proposed control scheme will be guaranteed in the sense that all the states and signals are uniformly bounded and arbitrary small attenuation level.
The first is the CTR in the presence of M-like node mobility (where M is an arbitrary bounded and obstacle free mobility model) and the second is the case of RWP mobility [15] (which is the most common mobility model used in the simulation of ad hoc networks).
convergence for random variables, are studied with the use of some properties of the geometry of projectors from U. The conditions are equivalent if (xn) is bounded, and are not equivalent for an arbitrary sequence.
Based on the results on finite-time boundned and average dwell time, sufficient conditions for finite-time bounded and finite-time H ∞ control under arbitrary switching are derived, and the closed-loop system trajectory stays within a prescribed bound.
Lastly, our approach easily generalizes to arbitrary geometries, both bounded and unbounded, and of varying genus.
Since is bounded in, and is arbitrary, by (2.28) we obtain a contradiction with.
t ∈ [ 0, T ] T. Hence, by the Arzelà-Ascoli theorem on time scales, { Q N x n } n = 1 ∞ and { K P, Q N x n } n = 1 ∞ are compact on an arbitrary bounded E ⊂ X, and the mapping N : X → Y is L-completely continuous.
J is a monotone and bounded operator in arbitrary Banach spaces; J is a strictly monotone operator in strictly convex Banach spaces; J is a continuous operator in smooth Banach spaces; J is a uniformly continuous operator on each bounded set in uniformly smooth Banach spaces; J is a bijection in smooth, reflexive, and strictly convex Banach spaces; J is the identity operator in Hilbert spaces.
It is easy to see that (35) is always solvable and, according to Corollary 5, the analyzed impulsive problem has bounded solution for arbitrary if the pulse parameter should be chosen as follows: (36).
