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Matrices and Linear Algebra
Published in William S. Levine, The Control Handbook: Control System Fundamentals, 2017
Various norms turn out to be appropriate for applications involving vectors in other vector spaces; as an example, a suitable norm for C[0, T], the space of real-valued continuous functions on the interval 0 ≤ t ≤ T is the uniform norm:
Fixed-time and finite-time stability of switched time-delay systems
Published in International Journal of Control, 2022
Notations: Let () be the set of (nonnegative) real numbers. The Euclidean norm of a vector or an absolute value of a real is denoted by . For a continuous function , , its uniform norm is defined as ; the space of such functions we will denote as . A continuous function belongs to class if it is strictly increasing and , and if it belongs to the class and it is also unbounded. A continuous function belongs to class if belongs to class and is strictly decreasing to zero for any fixed . The natural exponent .
Periodic solutions for partial neutral non densely differential equations
Published in Applicable Analysis, 2021
To put our problem into the abstract form, we set , the Banach space of continuous functions on , equipped with the uniform norm topology and we define by Using [34], we have Hence, is satisfied. Moreover, and we have, by [35], the part of A given by generates a -semigoup on satisfying We now consider the family defined on by for . Clearly that is 1-periodic, which means that is satisfied. Moreover, generates an evolution family defined by, Hence, using the superposition principle and (47), generates an evolution family on , defined by and satisfying That is, the evolution family has an exponential dichotomy with and constants such that for all , which is hypothesis .