We consider the problem of analytical synthesis of control for a nonlinear mathematical model, ensuring the motion of a part of the phase vector coordinates along a given spatial trajectory. A control algorithm is proposed that guarantees asymptotic approach and retention of a part of the phase vector components of a nonlinear system on a desired trajectory of motion. This trajectory is defined in terms of a certain curve in the spatial coordinate system.

We consider the transfer to a given orbit of a controlled spacecraft whose dynamics is described by the mathematical model of motion in a gravitational field with one attractive center. We propose a class of engine-thrust positional controls dependent on the problem parameters that solve the controllability problem for a given target orbit. The trajectory of motion of a phase point under such control is determined on an infinite time axis. On the initial time interval, the trajectory reaches a neighborhood of the target set, and thereafter it moves in this neighborhood asymptotically approaching the target. The positional control is derived in analytical form. Calculation results are reported for the positional control and the trajectories of motion using different test parameters and different types of spacecraft target trajectories.

The interaction of healthy and cancerous cell concentrations in diseases associated with blood cancer is described by a two-dimensional Lotka–Volterra competition model. A differential equation specifying the change in the concentration of a chemotherapeutic drug during treatment is added to the model. This equation includes a bounded control function determining the rate at which such a drug enters the patient’s bloodstream. The effectiveness of the used treatment is described by a nonmonotone therapy function. The problem is to minimize the weighted difference between the concentrations of cancerous and healthy cells at the end of a given treatment period for the considered three-dimensional control system. Application of the Pontryagin maximum principle allows to analytically study the properties of the optimal control. We single out and investigate possible cases when such a control is a bang-bang function and also the cases when, along with the bang-bang sections, it can contain singular regimes of the first and second orders. The established analytical results are confirmed by numerical calculations performed for different values of parameters and initial conditions of the considered minimization problem.

The Lotka–Volterra competition model is applied to describethe interaction between the concentrations of healthy and cancerous cellsin diseases associated with blood cancer. The model is supplemented witha differential equation characterizing the change in the concentration ofa chemotherapeutic drug. The equation contains a scalar bounded controlthat specifies the rate of drug intake. We consider the problem ofminimizing the weighted difference between the concentrations of cancerousand healthy cells at the end time of the treatment period. The Pontryaginmaximum principle is used to establish analytically the propertiesof an optimal control. We describe situations in which the optimal controlis a bang–bang function and situations in which the control may containa singular arc in addition to bang–bang arcs. The results obtained areconfirmed by corresponding numerical calculations.

ТРУДЫ ИНСТИТУТА МАТЕМАТИКИ И МЕХАНИКИ УрО РАН Том 27 № 2 2021 УДК

A model of production funds acquisition, which includes two differential links of the zero order and two series-connected inertial links, is considered in a one-sector economy. Zero-order differential links correspond to the equations of the Ramsey model. These equations contain scalar bounded control, which determines the distribution of the available funds into two parts: investment and consumption. Two series-connected inertial links describe the dynamics of the changes in the volume of the actual production at the current production capacity. For the considered control system, the problem is posed to maximize the average consumption value over a given time interval. The properties of optimal control are analytically established using the Pontryagin maximum principle. The cases are highlighted when such control is a bang-bang, as well as the cases when, along with bang-bang (non-singular) portions, control can contain a singular arc. At the same time, concatenation of singular and non-singular portions is carried out using chattering. A bang-bang suboptimal control is presented, which is close to the optimal one according to the given quality criterion. A positional terminal control is proposed for the first approximation when a suboptimal control with a given deviation of the objective function from the optimal value is numerically found. The obtained results are confirmed by the corresponding numerical calculations.

УДК 517.977.1 MSC: 49J15, 58E25, 92D25 DOI: 10.21538/0134-4889-2021-27-3-43-58 Рассматривается управляемая математическая модель лечения лейкемии, в основе которой лежит трехмерная модель хищник — жертва Лотки — Вольтерры. Эта модель описывает недавно разработанную технологию лечения лейкемии, представляющую собой терапию Т-клетками с химерными антигенными рецепторами (CAR-T

ТРУДЫ ИНСТИТУТА МАТЕМАТИКИ И МЕХАНИКИ УрО РАН

To the article “Optimal Control Problems for a Mathematical Model of the Treatment of Psoriasis,” by N. L. Grigorenko, É. V. Grigorieva, P. K. Roi, and E. N. Khailov, Vol. 30, No. 4, pp. 352–363, October, 2019.

We consider a terminal control problem for an underactuated nonlinear control system with phase constraints. The control is sought by the linearization method in which the control is determined by solving the Cauchy problem for a nonlinear parametric system of auxiliary differential equations. An auxiliary extremum problem with boundary conditions of general type is proposed for the numerical calculation of the control parameters. Examples of control are computed for a process with test parameters.

Differential game of two players with dynamics of the motion of the first player described by the second order equation and the second player controls the movement of the target point is considered. The coordinates of the target point become known at the current time. Conditions are proposed for the parameters of the game under which there is a control first player guaranteeing the end of the game in a finite time. The results of numerical calculations of controls and trajectories for the model parameters of the problem are presented.

We consider a nonlinear model of motion of a solid body with deficiency of control parameters. The model contains a disturbance parameter. We propose an open-loop control that takes the system from a given initial state to a given terminal state. Results of numerical calculations are presented for the dynamics of the components of the phase vector and of the controls.

We construct a mathematical model of quadcopter flight and design a positional control that solves the terminal control problem in the presence of a disturbance parameter. The study relies on the results of [6–9] and provides sufficient existence conditions for a positional control that solves the control problem for a class of boundary conditions.

We consider a terminal control problem with state constraints and additional constraints on the qualitative behavior of the terminal trajectory for a second-order system in two-dimensional Euclidean space under geometric constraints on control parameters. A class of control functions solving this control problem is proposed. Numerical results for the control system with model parameters are presented.

model of a two-dimensional open-pit mine is proposed and an optimal control problem is formulated with mixed constraints on the control parameters and an integral objective functional. The problem is discretized in one of the phase variables and solved by the gradient projection method with penalty functions. Numerical results illustrating the the method are also represented.

The article is devoted to the description of Academician Arkady Kryazhimskiy's life path. The facts of the scientific biography of Acad. Kryazhimskiy are presented with the emphasis on his outstanding contribution into the theory of dynamic inversion, the theory of differential games, and control theory. His personal talents in different spheres are also marked out.

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.

For a linear control system with constrained control, the problem of terminal control to a target point is considered. The starting point of the process belongs to a known set, but there is no information on which point of the set is the starting point. Sufficient conditions are given for the existence of a solution of the problem in the class of Yu.S. Osipov and A.V. Kryazhimskii's guaranteeing program packages. Calculation results are presented for a model example.