We obtain a representation of a solution of the Cauchy problem for a linear inhomogeneous differential equation with constant coefficients and pure delay. We use special matrix functions called a delayed matrix sine and a delayed matrix cosine. They have the form of matrix polynomials of degree dependent on the value of delay.

Systems are considered related to the control of processes described by oscillating second-order systems of differential equations with a single delay. An explicit representation of solutions with the aid of special matrix functions called a delayed matrix sine and a delayed matrix cosine is used to develop the conditions of relative controllability and to construct a specific control function solving the relative controllability problem of transferring an initial function to a prescribed point in the phase space.

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