The crucial benefit in the above formulation is the uniform bound on the leading operator [A.sub.c]
This can be easily seen by using a classical fixed point argument in Duhamel's formula ( ) together with the essential uniform bound (2.10) and (2.11).
Thanks to the essential bound (2.9) uniform bounds also hold on the derivatives ([mathematical expression not reproducible]).
Diverse initial states result in varied settling times, which means no uniform bound of the convergence time can be obtained in advance without the knowledge of maximum V([x.sub.0]).
Unlike existing works, the designed smooth fixed-time-convergent sliding mode controller motivates missile control variables to converge to the equilibrium point before the uniform bounded settling time in the presence of aerodynamic uncertainties with its input inherently continuous without using any discrete items, like the signum function.
Under assumptions of (2)-(5), for some M = M(U) > 0, one has the uniform bound
They derive the uniform bound
results under decay condition, which reads as follows:
The 15 papers included consider such topics as the Plucker-Clebsch formula in higher dimension, polynomial vector fields with algebraic trajectories, uniform bounds
for Hilbert coefficients of parameters, a property of the Frobenius map of a polynomial ring, and some homological properties of almost Gorensein rings.
2) In , for |a| = 1, the author was able to show the existence of a periodic solution under the stringent condition that the functions f and g are uniform bounded
by certain positive constants.
Given any sequence [M.sub.k] = ([M.sub.k], [G.sub.k], [Q.sub.k], [q.sub.k]) as before such that we have uniform bounds on the Riemannian curvature tensors [RM.sub.k] and their [p.sup.th] covariant derivatives
We also have uniform bounds on the curvatures of the [Mathematical Expression Omitted] and their derivatives, independent of k.