Том 27, № 3 (2016)

The article examines a controlled process of biological treatment of effluents. The model is described by a nonlinear system of three differential equations with one bounded control. We investigate the properties of the solutions of this system: boundedness and continuation on a given interval. The corresponding reachable set is considered. A bound is derived on the number of control switching points required to reach the set boundary.

We consider a class of nonlinear game-theoretical control problems for a dynamical system with uncertain parameters, a free termination time, and a Bolza functional in the form of a power function of the control parameter. A control is proposed in the class of countercontrols on which the objective functional attains a guaranteed value that depends on the realized uncertainty. Calculation results using test model parameters are reported for the control and the functional values.

For a certain class of two-dimensional bilinear controlled systems, we derive efficient sufficient conditions that ensure bang bang behavior of time-optimal controls.

We investigate the application of the Real Options approach to the optimization of open-pit mining. The Real Options approach introduces investment as an additional control parameter for profit maximization. In the context of applying the Real Options approach to open-pit mining optimization, we consider a model with two-stage investments. Open-pit mining requires both extracting and processing capacities. These capacities in turn require investments, which are divided into two parts: investments to create the initial capacities and investments to increase existing capacities in the process of mining. The initial and augmented capacities as well as the capacity augmentation time are control parameters that can be chosen with the objective of increasing the mining profits. In this article, we assume that the market price of the mineral is a random process described by a stochastic differential equation. A control strategy is a rule that at every time instant, making use of the available information, determines the mining rate, establishes if additional investments are required at the given time, and if yes, calculates the investment amount. The problem involves the construction of an optimal mining control strategy that maximizes the mean discounted profit from the open-pit mine.

We investigate modified models of information diffusion (or propagation) in a social group. The process dynamics is described by a one-dimensional controlled Riccati differential equation. The models in this article differ from the original model [2] by the choice of the optimand functional. Two variants of the optimand functional are considered. The optimal control problems are solved by the Pontryagin maximum principle [1]. We show that the optimal control program is a bang bang function of time with at most one switching point. Easily checked conditions on the problem parameters are derived, guaranteeing the existence of a switching point in the optimal control. Our theoretical analysis of the problem leads to the construction of a one-dimensional convex minimization problem to find the optimal control switching point. We also describe an alternative approach (without invoking the maximum principle) for the construction of the optimal solution that utilizes a special representation of the optimand functional and analyzes the reachable sets independent of the functional.