# The Oskolkov Equations on the Geometric Graphs as a Mathematical Model of the Traffic Flow

G.A. Sviridyuk, S.A. Zagrebina, A.S. KonkinaCurrently there arose a necessity of creation of adequate mathematical model describing the flow of traffic. The mathematical traffic control theory is now actively developing in the works of A.B. Kurzhanski and his school, where the transport flow is considered to be similar to the flow of an incompressible fluid, and consequently the hydrodynamic model, for example based on the system of Navier - Stokes Equations, is used. In addition to the obvious properties of traffic flow covered previously, such as viscosity and incompressibility, the authors of this article propose to take into consideration its elasticity. Indeed, when you turn on a forbidding signal of a traffic light vehicles do not stop instantly and smoothly reduce their speed up to stop accumulating before the stop line. Similarly, if you turn on an allowing signal of the traffic light vehicles do not start instantaneously and simultaneously, they start driving one after another, gradually raising up the speed. Thus the transport flow has an effect of retardation, which is typical for viscoelastic incompressible fluids described by a system of Oskolkov equations.Full text

The first part of the article substantiates a linear mathematical model, i.e. the model without convective terms in the Oskolkov equations. In the context of the model this means that transposition of vehicles can be neglected. In the second part the model is investigated on a qualitative level, i.e. we formulate the existence of a unique solution theorem for the stated problem and provide an outline of its proof.

- Keywords
- Oskolkov equation; geometric graph; Cauchy problem; traffic flows.
- References
- 1. Kurzhanski A.B. The Current Problems of the Dynamics and Control Theory, Motivation Theory and Computation. Road map [electronic resource]: plenary lecture at the meeting General Plenary. XII Russian Conference on Control, Moscow, Russia IPU RAN, 16 - 19 june 2014. Access mode: http://vspu2014.ipu.ru/conference/section_meeting_pubs?target=7860. - 09.07.2015 (in Russian)

2. Gasnikov A.V., Klenov S.L., Nurminski E.A., Kholodov Ya.A. etc. Vvedenie v matematicheskoe modelirovanie transportnykh potokov [Introduction to the Mathematical Modelling of Traffic Flows]. Moscow, MIPT, 2010. 362 p.

3. Oskolkov A. P. Some Nonstationary Linear and Quasilinear Systems Occurring in the Investigation of the Motion of Viscous Fluids. Journal of Soviet Mathematics, 1978, vol. 10, no. 2, pp. 299-335.

4. Pokornyi Yu.V., Penkin O.M., Pryadiev V.L. Differential Equations on Geometrical Graphs. Moscow, FizMatLit, 2004. (in Russian).

5. Sviridyuk G.A., Shemetova V.V. The Phase Space of a Nonclassical Model. Russian Mathematics (Izvestiya VUZ. Matematika), 2005, vol. 49, no. 11, pp. 44-49.

6. Sviridyuk G.A., Fedorov V.E. Linear Sobolev Type Equations and Degenerate Semigroups of Operators. Utrecht; Boston; Koln; Tokyo, VSP, 2003.

7. Zagrebina S.A., Soldatova E.A., Sviridyuk G.A. The Stochastic Linear Oskolkov Model of the Oil Transportation by the Pipeline. Semigroups of Operators - Theory and Applications. [International Conference], Bedlewo, Poland, Oktober 2013. Heidelberg; New York; Dordrecht; London: Springer Int. Publ. Switzerland, 2015, pp. 317-325,(Springer Proceedings in Mathematics & Statistics; vol. 113).

8. Manakova N.A. Optimal Control Problem for the Sobolev Type Equations. Chelyabinsk, Publ. Center of the South Ural State University, 2012. 88 p. (in Russian)

9. Sagadeeva M.A. Dichotomy of Solutions of Linear Sobolev Type Equations. Chelyabinsk, Publ. Center of the South Ural State University, 2012. 107 p. (in Russian)

10. Zamyshlyaeva A.A. Linear Sobolev Type Equations of High Order. Chelyabinsk, Publ. Center of the South Ural State University, 2012. 107 p. (in Russian)

11. Keller A.V. Chislennoe issledovanie zadach optimal'nogo upravleniya dlya modeley leont'evskogo tipa [Numerical Reseach of Optimal Control Problem for Leontieff Type Models. The Dissertation for Scientific Degree of the Doctor of Physical and Mathematical Sciences]. Chelyabinsk, South Ural State University, 2011. 252 p.

12. Shestakov A.L., Keller A.V., Nazarova E.I. The Numerical Solution of the Optimal Demension Problem. Automation and Remote Control, 2011, vol. 73, no. 1, pp. 97-104. DOI: 10.1134/S0005117912010079

13. Shestakov A. L. , Sviridyuk G. A., Butakova M. D. The Mathematical Modelling of the Production of Construction Mixtures with Prescribed Properties. Bulletin of the South Ural State University. Series: Mathematical Modelling, Programming & Computer Software (Bulletin SUSU MMCS), 2015, vol. 8, no. 1, pp. 100-110.

14. Oskolkov A. P. Nonlocal Problems for Some class Nonlinear Operator Equations Arising in the Theory Sobolev Type Equations. Journal of Soviet Mathematics, 1993, vol.64, no. 1, pp. 724-736.

15. Favini A., Lorenzi A., Tanabe H. First Order Regular and Degenerate Identification Differential Problems. Abstract and Applied Analysis, 2015. Article ID 393624, 42 pages, http://dx.doi.org/10.1155/2015/3936.