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.
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.
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.
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.
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
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.
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.
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.
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