# Developing industries in cooperative interaction: equilibrium and stability in processes with lag

### Abstract

A mathematical model of dynamic interaction between mining and processing industries is formalized and studied in the paper. The process of interaction is described by a system of two delay dierential equations. The criterion for asymptotic stability of nontrivial equilibrium point is obtained when both industries co-work steadily. The problem is reduced to nding stability criterion for quasi-polynomial of second order. Time intervals between deliveries of raw materials which make it possible to preserve stable interaction between the two industries are found.### References

X. Grigorieva and O.A. Malafeev, Competitive many-period postman problem with varying parameters, Applied Mathematical Sciences, vol. 8, no. 145-148, pp. 7249–7258, 2014.

A. Kirjanen and V. Veselov, The influence of capital turnover time and coefficient criteria of asymptotic stability of two competing firms, Proceedings of V International Conference State and Business, pp. 50-52, 2013.

A. Kirjanen and O. Shalyapina, Pointwise synthes controllability of systems with delay, Contemporary Engineering Sciences, vol. 2, pp. 14–18, 1989.

J. K. Hale, Theory of functional differential difference equations, Springer Verlag, New York, 1977.

A. I. Kirjanen, Stability of systems with aftereffect and there applications, St.Petersburg State University, Russia, Saint-Petersburg, 1994.

A.Kirjanen, Coefficient criteria of stable coexistence of two competitors, Stability and Control Processes 2015 - Proceedings, pp.463–466, 2015.

L.A. Bondarenko and A.V. Zubov and V.B. Orlov and V.A. Petrova and N.S. Ugegov, Application in practice and optimization of industrial information systems, Journal of Theoretical and Applied Information Technology, vol. 85, no.3, pp. 305–308, 2016.

A.V.Zubov and A.Y. Murashko and L.G. Kolyada and E.A. Volkova and O.A. Zubova, Fidelity issue of engineering analysis and computer aided calculations in sign models of dynamic systems, Global Journal of Pure and Applied Mathematics, vol. 12, no.5, pp. 4203-4217, 2016.

A.I. Kirjanen and A.A. Samodurov, Influence of external factors on business process of company in dynamics, Proc. of the 7th International Research and Practical Conference, pp. 88–92, 2015.

A.I.Kirjanen and A.V.Labudin and A.A.Samodurov, Business-Process Approach to Economic Security Management of the Enterprise with Attraction of the Equation of Profitability, Russian Public Policy Journal (RuPP) Upravlenceskoe konsultirovanie, vol. 3, pp.96–105, 2016.

V.N.Kolokoltsov and O.A.Malafeyev, Mean-Field-Game Model of Corruptions, Dynamic Games and Applications, vol. 7, no.1, pp.34–47, 2017.

O.A. Malafeyev and N.D. Redinskikh and G.V. Alferov, Electric circuits analogies in economics modeling: Corruption networks, BDO 2014 - Proceedings, http://dx.doi.org/10.1109/Emission.2014.6893965, 2014.

O.A. Malafeyev and L.A. Petrosyan, Differential search games - Dynamic-games with complete information Vestnik Leningradskogo Universiteta, Seriya Matematika, Mekhanika, Astronomiya, vol. 2, pp. 26-30, 1983.

O.A. Malafeev, Existence of equilibrium points in differential noncooperative many-person games, Vestnik Leningradskogo Universiteta, Seriya Matematika, Mekhanika, Astronomiya, vol.3, pp.40-46, 1982.

O.A. Malafeev, On the existence of equilibrium points in 2-person differential games with separated dynamics, Vestnik Leningradskogo Universiteta, Seriya Matematika, Mekhanika, Astronomiya, pp.12-16, 1980.

V.V. Malygina and M.M. Kipnis, The stability cone for a matrix delay difference equation, International Journal of Mathematics and Mathematical Sciences, http://dx.doi.org/10.1155/2011/860326, 2011.

E.G. Neverova and O.A. Malafeyef, A model of interaction between anticorruption authority and corruption groups, AIP Conference

Proceedings, http://dx.doi.org/10.1063/1.4912671, 2015.

Yu.A. Pichugin and O.A. Malafeev, Statistical estimation of corruption indicators in the firm, Applied Mathematical Sciences, vol.10, no.42, pp.2065-2073, 2016.

L.S. Pontryagin, On the zeros of some elementary transcendental functions, American Mathematical Society, vol.2, pp. 95–110,1955.

Y.V.Kozachenko and I.V.Rozora, A criterion for testing hypothesis about impulse response function, Statistics, Optimization and Information Computing, vol. 4, pp. 214–232, 2016.

O.Kostyukova and T.Tchemisova and M.Kurdina, On optimal properties of special nonlinear and semi-infinite problems arising in parametric optimization, Statistics, Optimization and Information Computing, vol. 5, pp. 99-108, 2017.

N.N. Subbotina and E.A. Kolpakova, Method of Characteristics for Optimal Control Problems and Conservation Laws, Journal of Mathematical Sciences, vol. 199, no.5, pp. 588–595, 2014.

M.Luz and M.Moklyachuk, Filtering problem for functionals of stationary sequences, Statistics, Optimization and Information Computing, vol.4, pp.68–83, 2016.

O.A.Malafeev and S.A.Nemnyugin, Generalized dynamic model of a system moving in an external field with stochastic components, Theoretical and Mathematical Physics, vol.107, pp.770–774, 1996.

E.G.Neverova and O.A.Malafeyev and G.V.Alferov and T.E.Smirnova, Model of interaction between anticorruption authorities and corruption groups, Stability and Control Processes 2015 - Proceedings, pp.488–490, 2015.

G.V.Alferov and O.A.Malafeyev, The robot control strategy in a domain with dynamical obstacles, Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics), vol.1093, pp.211–217,1996.

O.A. Malafeev, Stationary strategies in differential games, USSR Computational Mathematics and Mathematical Physics, vol.17,pp.37–46, 1977.

O.A.Malafeev, Equilibrium situations in dynamic games, Cybernetics, vol.10, pp.504–513, 1974.

Z.Taushanov and A.Berchtold, A direct local search method and its application to a markovian model, Statistics, Optimization and Information Computing, vol. 5, no. 1, pp. 19-34, 2017.

Y.V.Kozachenko and M.Y.Petranova, Proper complex random processes, Statistics, Optimization and Information Computing, vol.5, no. 2, pp. 137-146, 2017.

M.Moklyachuk and M.Sidei, Filtering problem for stationary sequences with missing observations, Statistics, Optimization and Information Computing, vol. 4, no. 4, pp. 308-325, 2016.

N. Xiu, A SQP method for general nonlinear complementarity problems, Applied Mathematics, vol.15, no.4, pp. 433-442, 2000.

N. Xiu, A class of trust region methods for linear inequality constrained optimization and its theory analysis: I. Algorithm and global convergence, Applied Mathematics, vol.10, no.3, pp. 287-296, 1995.

O.A.Malafeyev and S.A.Nemnyugin and G.A.Ivaniukovich, Stochastic models of social-economic dynamics, Stability and Control Processes 2015 - Proceedings, pp.483–485, 2015.

G.D.Drozdov and O.A.Malafeyev and S.A.Nemnyugin, Multicomponent dynamics of competitive single-sector economy development, Stability and Control Processes 2015 - Proceedings, pp.457–459, 2015.

O.A.Malafeev, The existence of situations of "-equilibrium in dynamic games with dependent movements, USSR Computational Mathematics and Mathematical Physics, vol.14, pp.88–99, 1974.

*Statistics, Optimization & Information Computing*,

*5*(4), 341-347. https://doi.org/10.19139/soic.v5i4.357

- Authors retain copyright and grant the journal right of first publication with the work simultaneously licensed under a Creative Commons Attribution License that allows others to share the work with an acknowledgement of the work's authorship and initial publication in this journal.
- Authors are able to enter into separate, additional contractual arrangements for the non-exclusive distribution of the journal's published version of the work (e.g., post it to an institutional repository or publish it in a book), with an acknowledgement of its initial publication in this journal.
- Authors are permitted and encouraged to post their work online (e.g., in institutional repositories or on their website) prior to and during the submission process, as it can lead to productive exchanges, as well as earlier and greater citation of published work (See The Effect of Open Access).