The problems of controllability of linear processes with time delays described by the differential equation <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">ẋ(t) = A(t)x(t) + ∑min{i=0}max{k} Bᵢ(t)u(t-hᵢ</tex> and satisfied in Euclidean space R n are considered. The necessary and sufficient conditions for controllability, using the transition matrix of the process, are given. By the additional assumption that the matrices <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">A,Bᵢ</tex> are analytic in <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">t</tex> , another criterion of controllability, using the matrices <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">A</tex> and B <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">i</inf> only, is proved.
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A.W. Olbrot (1972) studied this question.