ISSN 0012-2661, Differential Equations, 2019, Vol. 55, No. 1, pp. 113–119. c  Pleiades Publishing, Ltd., 2019. Russian Text c  N.G. Pavlova, 2019, published in Differentsial’nye Uravneniya, 2019, Vol. 55, No. 1, pp. 111–116. CONTROL THEORY Study of the Continuous-Time Open Dynamic Leontief Model as a Linear Dynamical Control System N. G. Pavlova 1,2 1 Trapeznikov Institute of Control Sciences of the Russian Academy of Sciences, Moscow, 117997 Russia 2 Peoples’ Friendship University of Russia, Moscow, 117198 Russia e-mail: natasharussia@mail.ru Received June 17, 2018 Abstract—We study the topological properties of the technology set (the set of all technologi- cally admissible net production output vectors) in the continuous-time dynamic Leontief model in which the control is a function of unproductive consumption. Necessary conditions for the technology set to be closed are obtained. DOI: 10.1134/S0012266119010117 INTRODUCTION The Leontief model is a mathematical model of economics characterizing the relationship be- tween the output in one branch of industry and the expenditures of all branches participating in the manufacturing process necessary to ensure this output. Leontief began to develop the theoretical foundations of this model as early as in 1923 [1, 2]. The first dynamic interindustry models of national economics appeared in the 1970s [3–5]. In the present paper, the continuous-time open dynamic Leontief model is viewed as a linear dynamical control system. For the industrial model in question, we study the topological properties of the technology set. (The technology set is the set of all technologically admissible net production output vectors). The entries of the direct expenditure matrix in the dynamic model contain the direct material expenditures and the fixed asset disposal and repair costs, and the amount of capital investments depends on the gross output according to Harrod’s linear acceleration principle. In the model under study, the control is a function of the unproductive consumption and ranges in a finitely generated convex cone. The model can be used to study the behavior of an economical system on long time intervals under the assumption that there is no technical progress. In the model under study, the production process is described in terms of the technology set. One economically important property of the technology set is that it is closed; i.e., one can use the “extreme” operation modes. In the present paper, we obtain necessary conditions for the technology set to be closed in the continuous-time open dynamic Leontief model. 1. AUXILIARY RESULTS Consider a plant whose behavior is described by the linear system of ordinary differential equa- tions ˙ x = Ax + Bu, t ∈ [0,T ], (1) with the initial condition x(0) = 0. (2) 113