# The Optimal Control of the Behavior of Enterprises of Cellular Communication in the Conditions of Competitive Struggle

Irina Bolodurina^{*} and Tatyana Ogurtsova

Mathematical modelling is important issue of analysis of mobile operators behaviour competing for shared sources. Consider n competing firms providing cellular communication services existing in same economic field i. e. with joint labour costumer and natural sources. To develop optimal control model of cellular communication companies behaviour divide all operators on two unequal groups: the economic agent 1(EA1) means leading enterprise of this industry and its competitors - economic agent 2 (EA2), combining the rest of the enterprise by summing the number of subscribers in the market. Denote $x_i(t)$ number of subscribers of i-th economic agent at moment $t$, $x_{i0}$ -- number of subscribers of i-th economic agent at initial time moment, $(i=1,2)$.\newline To take into account exponential growth of number of subscribers if no competitors, non-linear character of interaction, lag and competitors actions, use logistic lag model (1). \begin{equation} \dot{x}(t)=x_i \left[ \varepsilon_i - \sum \limits _{k=1}^2 \gamma_{ik}x_k(t-\tau) \right], \end{equation} where $\gamma_{ik}, \ k=1,2$ -- are the coefficients of $i$-th agent impact on $k$-th agent. Number of subscribers of the $i$-th economic agent for the initial time interval $[-\tau,0]$ is $\varphi_i(t),$ $i=1,2$ \begin{equation} x_i(t)=\varphi_i(t), \ t\in [-\tau,0]. \end{equation} The existence of the $\eta_i$ lower bound of the volume of the subscriber base, which ensures the normal functioning of the enterprise, and also the upper bound $\mu_i$ defined by the technical specifications of the network, i.e. the possibility to maintain a specified number of subscribers per unit of time, described by the inequalities (3) \begin{equation} \eta_i \leq x_i(t) \leq \mu_i, \ i=1,2. \end{equation} An approach to the simultaneous identification of the lag and the coefficients of the system, which is based on a method to set up a model on experimental data [5]. To demonstrate the importance of the introduction of lag we consider identifying of parameters without this delay. Competition between companies is for the potential customer who is interested in the cost of a minute of communication services. Therefore there is a problem of efficient control of the companies pricing. Let $u_i(t), i=1,2$ an indicator of average cost per minute that satisfy the constraint (4). \begin{equation} \alpha \leq u_i(t) \leq \beta, \ t \in [0,T], \end{equation} where $\alpha$ -- is the minimum average cost per connection minute; $\beta$ -- maximum average cost per connection minute, allowing economic agents to remain competitive in the market.\\ As the structure of the model we select a dynamic model as a system of lag differential equations type models Lotka-Volterra (5). \begin{equation} \begin{matrix} \dot{x}_1(t)=x_1(t)\left[ \varepsilon_1-\gamma_{11}x_1(t-\tau)-\gamma_{12}x_2(t-\tau)\right] - p_{11}u_1(t)-p_{12}u_2(t), \\ \dot{x}_2(t)=x_2(t)\left[ \varepsilon_2-\gamma_{21}x_1(t-\tau)-\gamma_{22}x_2(t-\tau)\right] - p_{21}u_1(t)-p_{22}u_2(t), \end{matrix} \end{equation} where $p_{ij}, \ i,j=1,2$ -- are the coefficients of communication minute average cost impact on the increase of the subscribers number. \\ In the process of economic agents interaction the company may deliver next criteria of quality: \begin{enumerate} \item to build the subscriber base of the enterprise for a finite period of time \begin{equation} J_1(u_1)=\int \limits_0 ^T b_1x_1(t)dt \to \max; \end{equation} \item to bring the subscriber base of the enterprise for the given volume in the final moment of time \begin{equation} J_2(u_1)=b_2\left( x_1(T)-M\right)^2 \to \min, \end{equation} where $M$ -- planning value of the subscriber base EA1; \item to increase of the profit of the enterprise for a finite period of time \begin{equation} J_3(u_1)=\int \limits_0 ^T b_3x_1(t)u_1(t)dt \to \max. \end{equation} \end{enumerate} Depending on the priorities of the development of agent EA1 solve the problem of optimal control for each of the functional (6) - (8). These problems belong to a class of optimal control problems with constant lag. This solutions applied the Pontryagin maximum principle for systems with constant lag on the assumption that the value of the EA2 communication minute cost is fixed, and can be estimated from the dynamics of the previous pricing and market trends. \\ Implementation of the phase constraint is ensured through the introduction of the quadratic penalty external components in these functions.\\ Features built optimal control problem is the non-linearity of the differential equations system describing the dynamics of the development of subscriber base of two economic agents, as well as the presence of a constant lag in the managed system state vector. Taking into account the peculiarities for the constructed problem of optimal control of necessary optimality conditions are obtained on the basis of which implemented the numerical algorithm of solving it. Discrete approximation of continuous model was built. The constraint on the control is available by the method of gradient projection at the arbitrary choice of the initial approximation management. \\ It should be noted that the numerical method is not always possible to find a solution of the optimal control problem with a required accuracy. Therefore, in the process of problem solving can be consistently implemented various algorithms. One of approximate methods for solving optimal control of non-linear objects offered by L.I. Shatrovsky. It is based on the linearisation of a given non-linear system and further iterative procedure, during which when given as a function of the time of the initial approximation control on each step is solved linear objective of approximating the original problem. To improve the reliability of the optimal control calculation for nonlinear problems, authors has proposed the combined method, the essence of initial approximation by the projection of gradient method with valid management selected obtained in the method Shatrovsky. This approach to the initial approximation of management will allow to avoid functionality in the local extremum. In the problem has developed a software package with the programming environment Borland Delphi 7.0, implements the numerical decision of problems of optimal control of the communication enterprises behaviour in the conditions of competition for consumers. The program complex contains four modules: M1 -- main window module, M2 -- model parameters identification module, M3 is gradient projection method with an arbitrary choice of the initial approximation management, M4 -- projection of gradient method module with the choice of initial approximation based management method Shatrovsky. According to the available data of the subscriber base, the dynamics of the previous pricing and market trends allows us to find optimal values for the parameters and lag, as well as optimal management scenarios. \begin{thebibliography}{99} \bibitem{AT} Andreeva E.A., Tciruleva V.M., {\em Calculus of variations and optimization methods}, Orenburg, Tver: Tver state University, 2004. - 575. \bibitem{bolodurina} Bolodurina I.P. Differential equations with delay argument and applications: a tutorial Orenburg. st. un-ty, 2006. - 101 c. \bibitem{BO} Bolodurina I.P., Ogurtsova T.A., {\em Managing the price for services of the enterprises of the telecommunications industry. The Problems of management}. 2011. N3. P. 30-35. \bibitem{KS} Koblov A.I., Shiryaev V.I., {\em Optimal control of the behaviour of firms on the market of cellular communication}. Theory and systems. 2008. N5. P. 157-165. \bibitem{Prasolov} Prasolov A.V., {\em Dynamic models with delay and applications in Economics and engineering}, SPb.: Fallow deer, 2010. - 192 p. \end{thebibliography}

Mathematics Subject Classification: 49M05

Keywords: mathematical model; differential equations with lag; optimal control; identification of parameters; principal of Pontryaginâ€™s maximum

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