Journal
MATHEMATICS
Volume 11, Issue 4, Pages -Publisher
MDPI
DOI: 10.3390/math11040806
Keywords
stability; controllability; delay systems; fractional calculus
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In this paper, we introduce the delayed sine/cosine-type matrix function and use the Laplace transform method to find a closed form solution for the initial value problem (IVP) of a second-order time-delayed linear system with noncommutative matrices A and omega. We also introduce a delay Gramian matrix and study a relative controllability linear/semi-linear time delay system. We derive the necessary and sufficient condition for the relative controllability of the linear time-delayed second-order system and provide sufficient conditions for the relative controllability of the semi-linear second-order time-delay system. Finally, we investigate the Ulam-Hyers stability of a second-order semi-linear time-delayed system.
We introduce the delayed sine/cosine-type matrix function and use the Laplace transform method to obtain a closed form solution to IVP for a second-order time-delayed linear system with noncommutative matrices A and omega. We also introduce a delay Gramian matrix and examine a relative controllability linear/semi-linear time delay system. We have obtained the necessary and sufficient condition for the relative controllability of the linear time-delayed second-order system. In addition, we have obtained sufficient conditions for the relative controllability of the semi-linear second-order time-delay system. Finally, we investigate the Ulam-Hyers stability of a second-order semi-linear time-delayed system.
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