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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 <i>A</i> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi mathvariant="normal">Ω</mi></mrow></semantics></math></inline-formula>. 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.