Abstract
Dynamic interindustry balance models are described by differential equations of the first order, and the matrix at the derivative, which is a matrix of specific capital expenditures, can be degenerated. The stationary case, including optimal control problems, is well studied for such models. We consider the non-stationary case, where one of the matrices is multiplied by the scalar function that depends on time. In the stationary case, the results of the study of optimal control problems for degenerate balance interindustry models are presented by methods of the theory of degenerate matrix groups. Highlight that there is the importance of considering this problem for Leontief type models. Only for this particular case there is a convergence of numerical solutions to the optimal control problem to the exact one. We introduce the concept of flow of degenerate matrices and use them to construct the solutions of the non-stationary Leontief type system. We use these solutions in order to investigate an optimal control problem for non-stationary Leontief type systems of the specified type and we have proved the existence of a unique solution to the problem. The obtained results are illustrated by the example of solving an optimal control problem for the classical Leontief model in the non-stationary case.
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Keller, A.V., Sagadeeva, M.A. (2020). Degenerate Matrix Groups and Degenerate Matrix Flows in Solving the Optimal Control Problem for Dynamic Balance Models of the Economy. In: Banasiak, J., Bobrowski, A., Lachowicz, M., Tomilov, Y. (eds) Semigroups of Operators – Theory and Applications. SOTA 2018. Springer Proceedings in Mathematics & Statistics, vol 325. Springer, Cham. https://doi.org/10.1007/978-3-030-46079-2_15
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