A nonlinear controlled system of differential equations describing the process of production and sales of a consumer good
Ellina Grigorieva Evgenii Khailov
Conference Publications 2003, 2003(Special): 359-364 doi: 10.3934/proc.2003.2003.359
A nonlinear controlled system of differential equations has been constructed to describe the process of production and sales of a consumer good. This model can be controlled either by the rate of production or by the price of the good. The attainable sets of corresponding controlled systems are studied. It is shown that in both cases the boundaries of these sets are the unions of two two-parameter surfaces. It is proved that every point on the boundaries of the attainable sets is a result of piecewise constant controls with at most two switchings. Attainable sets for different values of parameters of the model will be demonstrated using MAPLE.
keywords: microeconomic dynamical model. Nonlinear controlled system attainable set
Optimal control of pollution stock
Ellina Grigorieva Evgenii Khailov
Conference Publications 2011, 2011(Special): 578-588 doi: 10.3934/proc.2011.2011.578
A model of an interaction between a manufacturer and the state where the manufacturer produces a single product and the state controls the level of pollution is created and investigated. The model is described by a nonlinear system of two differential equations with two bounded controls. The best optimal strategy is found analytically with the use of the Pontryagin Maximum Principle and Green’s Theorem.
keywords: nonlinear model environmental problem optimal control
Optimal control of a nonlinear model of economic growth
Ellina Grigorieva Evgenii Khailov
Conference Publications 2007, 2007(Special): 456-466 doi: 10.3934/proc.2007.2007.456
A nonlinear control model of a firm describing the change of production and accumulated $R&D$ investment is investigated. An optimal control problem with $R&D$ investment rate as a control parameter is solved. Optimal dynamics of economic growth of a firm versus the current cost of innovation is studied. It is analytically determined that dependent on the model parameters, the optimal control must be of one of the following types : a) piecewise constant with at most two switchings, b) piecewise constant with two switching and containing a singular arc. The intervals on which switching from regular to singular arcs occur are found numerically. Finally, optimal investment strategies and production activities are compared with econometric data of an actual firm.
keywords: model of economic growth. Optimal control
Optimal control of a commercial loan repayment plan
Ellina Grigorieva Evgenii Khailov
Conference Publications 2005, 2005(Special): 345-354 doi: 10.3934/proc.2005.2005.345
We consider a controlled system of differential equations modeling a firm that takes a loan in order to expand its production activities. The objective is to determine the optimal loan repayment schedule using the variables of the business current profitability, the bank's interest rate on the loan and the cost of reinvestment of capital. The portion of the annual profit which a firm returns to the bank and the value of the total loan taken by the firm are control parameters. We consider a linear production function and investigate the attainable sets for the system analytically and numerically. Optimal control problems are stated and their solutions are found using attainable sets. Attainable sets for different values of the parameters of the system are constructed with the use of a computer program written in MAPLE. Possible economic applications are discussed.
keywords: attainable set microeconomic dynamical model. Controlled system
Hierarchical differential games between manufacturer and retailer
Ellina Grigorieva Evgenii Khailov
Conference Publications 2009, 2009(Special): 300-314 doi: 10.3934/proc.2009.2009.300
A two-dimensional microeconomic model with three bounded controls is created and investigated. The model describes a manufacturer producing a consumer good and a retailer that buys this product in order to resell it for a profit. Two types of differential hierarchical games will be applied in order to model the interactions between the manufacturer and retailer. We will consider the difficult case in which the maximum of the objective functions can be reached only on the boundary of the admissible set. Optimal strategies for manufacturer and retailer in both games will be found. The object of our interest is the investigation of the vertical integration of retail and industrial groups. We will determine the conditions of interaction that produce a stable and maximally effective structure over given planning periods.
keywords: Economic Models Optimal Control Game Theory
Determination of the optimal controls for an Ebola epidemic model
Ellina Grigorieva Evgenii Khailov
Discrete & Continuous Dynamical Systems - S 2018, 11(6): 1071-1101 doi: 10.3934/dcdss.2018062

A control SEIR type model describing the spread of an Ebola epidemic in a population of a constant size is considered on the given time interval. This model contains four bounded control functions, three of which are distancing controls in the community, at the hospital, and during burial; the fourth is burial control. We consider the optimal control problem of minimizing the fraction of infectious individuals in the population at the given terminal time and analyze the corresponding optimal controls with the Pontryagin maximum principle. We use values of the model parameters and control constraints for which the optimal controls are bang-bang. To estimate the number of zeros of the switching functions that determine the behavior of these controls, a linear non-autonomous homogenous system of differential equations for these switching functions and corresponding to them auxiliary functions are obtained. Subsequent study of the properties of solutions of this system allows us to find analytically the estimates of the number of switchings and the type of the optimal controls for the model parameters and control constraints related to all Ebola epidemics from 1995 until 2014. Corresponding numerical calculations confirming the results are presented.

keywords: SEIR model control the spread of Ebola epidemic nonlinear control system Pontryagin maximum principle non-autonomous quadratic differential system generalized Rolle's theorem
Parametrization of the attainable set for a nonlinear control model of a biochemical process
Ellina Grigorieva Evgenii Khailov Andrei Korobeinikov
Mathematical Biosciences & Engineering 2013, 10(4): 1067-1094 doi: 10.3934/mbe.2013.10.1067
In this paper, we study a three-dimensional nonlinear model of a controllable reaction $ [X] + [Y] + [Z] \rightarrow [Z] $, where the reaction rate is given by a unspecified nonlinear function. A model of this type describes a variety of real-life processes in chemical kinetics and biology; in this paper our particular interests is in its application to waste water biotreatment. For this control model, we analytically study the corresponding attainable set and parameterize it by the moments of switching of piecewise constant control functions. This allows us to visualize the attainable sets using a numerical procedure.
    These analytical results generalize the earlier findings, which were obtained for a trilinear reaction rate (which corresponds to the law of mass action) and reported in [18,19], to the case of a general rate of reaction. These results allow to reduce the problem of constructing the optimal control to a straightforward constrained finite dimensional optimization problem.
keywords: waste water biotreatment Environmental problem nonlinear reaction rate attainable set.
An optimal control problem in HIV treatment
Ellina Grigorieva Evgenii Khailov Andrei Korobeinikov
Conference Publications 2013, 2013(special): 311-322 doi: 10.3934/proc.2013.2013.311
We consider a three-dimensional nonlinear control model, which describes the dynamics of HIV infection with nonlytic immune response and possible effects of controllable medication intake on HIV-infected patients. This model has the following phase variables: populations of the infected and uninfected cells and the concentration of an antiviral drug. The medication intake rate is chosen to be a bounded control function. The optimal control problem of minimizing the infected cells population at the terminal time is stated and solved. The types of the optimal control for different model parameters are obtained analytically. This allowed us to reduce the two-point boundary value problem for the Pontryagin Maximum Principle to one of the finite dimensional optimization. Numerical results are presented to demonstrate the optimal solution.
keywords: Valee-Poussin Theorem. nonlinear control system Pontryagin Maximum Principle HIV treatment
Optimal control for an epidemic in populations of varying size
Ellina Grigorieva Evgenii Khailov Andrei Korobeinikov
Conference Publications 2015, 2015(special): 549-561 doi: 10.3934/proc.2015.0549
For a Susceptible-Infected-Recovered (SIR) control model with varying population size, the optimal control problem of minimization of the infected individuals at a terminal time is stated and solved. Three distinctive control policies are considered, namely the vaccination of the susceptible individuals, treatment of the infected individuals and an indirect policy aimed at reduction of the transmission. Such values of the model parameters and control constraints are used, for which the optimal controls are bang-bang. We estimated the maximal possible number of switchings of these controls, which task is related to the estimation of the number of zeros of the corresponding switching functions. Different approaches of estimating the number of zeros of the switching functions are applied. The found estimates enable us to reduce the optimal control problem to a considerably simpler problem of the finite-dimensional constrained minimization.
keywords: non-autonomous quadratic di fferential system Valee- Poussin Theorem. generalized Rolle's Theorem control the spread of infection nonlinear control system SIR model Pontryagin maximum principle
Mathematical insights on psoriasis regulation: Role of Th1 and Th2 cells
Amit Kumar Roy Priti Kumar Roy Ellina Grigorieva
Mathematical Biosciences & Engineering 2018, 15(3): 717-738 doi: 10.3934/mbe.2018032

Psoriasis is an autoimmune disorder, characterized by hyper-proli-feration of Keratinocytes for the abnormal activation of T Cells, Dendritic Cells (DCs) and cytokine signaling. Interaction of DCs and T Cells enable T Cell to differentiate into Type 1 (Th1), Type 2 (Th2) helper T Cell depending on cytokine release. Hyper-proliferation of Keratinocytes may occur due to over expression of pro-inflammatory cytokines secreted by Th1-Cells viz. Interferon gamma ($\mbox{IFN}-{γ}$), Transforming growth factor beta ($\mbox{TGF}-β$) and Tumor necrosis factor alpha ($\mbox{TNF}-α$) etc. Deregulation of epidermal happens due to signaling of anti-inflammatory cytokines like Interleukin 10 ($\mbox{IL}-{10}$), Interleukin 4 ($\mbox{IL}-{4}$) etc., released by Th2-Cells. In this article, we have constructed a set of nonlinear differential equations involving the above cell population for better understanding the impact of cytokines on Psoriasis. System is analyzed introducing therapeutic agent (Biologic / $\mbox{IL}-{10}$) for reducing the hyper-proliferation of Keratinocytes. Effect of Biologic is used as a surrogate of control parameter to reduce the psoriatic lesions. We also studied its effect both in continuous and impulsive dosing method. Our study reveals that impulsive dosing is more applicable compare with continuous dosing to prevent Psoriasis.

keywords: Psoriasis Cytokines Keratinocytes Biologic Optimal control Impulsive approach

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