### Article

## Algorithmic method for modeling the optimal treatment of patients with HIV

The mathematical model describing the dynamics of HIV in the human body is a nonlinear system of differential equations. This model takes into account the effect of drugs on the body. Thus, it is possible to obtain ”optimal” treatment regimens for patients, which cause minimal harm to the body. In the work for constructing suboptimal control of the supply of drugs, the method of ”extended linearization” is used, which makes it possible to switch from a nonlinear model to a linear model, but with parameters that depend on the state. To solve the resulting equation Riccati and search for control actions, a method is proposed for the formation of optimization algorithms for nonlinear control systems based on the application of functions of admissible values of control actions.

17th IFAC Workshop on Control Applications of Optimization CAO 2018

Yekaterinburg, Russia, 15–19 October 2018

The problem of robust admissibilization with guar- anteed random disturbance attenuation level for discrete-time time-invariant (LDTI) descriptor systems with norm-bounded uncertainties is considered. The input disturbance is supposed to be a stationary Gaussian sequence with bounded mean anisotropy level. The solutions to analysis and robust state- feedback control problems are formulated in terms of matrix inequalities. The obtained conditions are convex over all decision variables and do not require inverse matrix searching. Compar- ison between iterative [1], [2] and non-iterative methods is given.

An optimal control problem is formulated for a class of nonlinear systems for which there exists a coordinate representation (diffeomorphism) transforming the original system into a system with a linear main part and a nonlinear feedback. In this case the coordinate transformation significantly changes the form of original quadratic functional. The penalty matrices become dependent on the system state. The linearity of the structure of the transformed system and the quadratic functional make it possible to pass over from the Hamilton–Jacoby–Bellman equation to the Riccati type equation with state-dependent parameters upon the control synthesis. Note that it is impossible to solve the obtained form of Riccati equation analytically in the general case. It is necessary to approximate the solution; this approximation is realized by numerical methods using symbolic computer packages or interpolation methods. In the latter case, it is possible to obtain the suboptimal control. The presented example illustrates the application of the proposed control method for the feedback linearizable nonlinear system.

**The problem of optimal control is formulated for a class of nonlinear objects that can be represented as objects with a linear structure and parameters that depend on the state. The linear structure of the transformed nonlinear system and the quadratic functional of quality allow for the synthesis of optimal control, i.e. parameters of the regulator, move from the need to search for solutions of the Hamilton-Jacobi equation to an equation of the Riccati type with parameters that depend on the state.** **The main problem of implementing optimal control is related to the problem of finding a solution to such an equation at the pace of object functioning.** **The paper proposes an algorithmic method of parametric optimization of the regulator. This method is based on the use of the necessary conditions for the optimality of the control system under consideration. The constructed algorithms can be used both to optimize the non-stationary objects themselves, if the corresponding parameters are selected for this purpose, and to optimize the entire managed system by means of the corresponding parametric adjustment of the regulators. The example of drug treatment of patients with HIV is demonstrated the effectiveness of the developed algorithms.**

This paper is dedicated to optimal state-feedback control problem for discrete-time descriptor systems in presence of “colored” noise with known mean anisotropy level. Here “colored” noise stands for a stationary Gaussian sequence, generated by a linear shaping filter from the Gaussian white noise sequence. The control goal is to find a state feedback control law which makes the closed-loop system admissible and minimizes its a-anisotropic norm (mean anisotropy level a is known).

The theoretical fundamentals for solving the linear quadratic problems may be some times used to design the optimal control actions for the nonlinear systems. The method relying on the Riccati equation with state-dependent coefficients is promising and rapidly developing tools for design of the nonlinear controllers. The set of possible suboptimal solutions is generated by ambiguous representation of the nonlinear system as a linearly structured system with state-depended coefficients and the lack of sufficiently universal algorithms to solve the Riccati equation also having state-depended coefficients. The paper proposed a method to design a guaranteed control for the uncertain nonlinear plant with state-depended parameters. An example of design the controller for an uncertain nonlinear system was presented.

This collection of articles contain materials of the talks presented at the International Conference "Systems Analysis: Modeling and Control" in memory of Academician A.V. Kryazhimskiy, Moscow, May 31 - June 1, 2018

We consider an optimal investment and consumption problem for a Black-Scholes financial market with stochastic volatility and unknown stock price appreciation rate. The volatility parameter is driven by an external economic factor modeled as a diffusion process of Ornstein- Uhlenbeck type with unknown drift. We use the dynamical programming approach and find an optimal financial strategy which depends on the drift parameter. To estimate the drift coefficient we observe the economic factor Y in an interval [0, T0] for fixed T0 > 0, and use sequential estimation. We show that the consumption and investment strategy calculated through this sequential procedure is δ-optimal.

A model for organizing cargo transportation between two node stations connected by a railway line which contains a certain number of intermediate stations is considered. The movement of cargo is in one direction. Such a situation may occur, for example, if one of the node stations is located in a region which produce raw material for manufacturing industry located in another region, and there is another node station. The organization of freight traﬃc is performed by means of a number of technologies. These technologies determine the rules for taking on cargo at the initial node station, the rules of interaction between neighboring stations, as well as the rule of distribution of cargo to the ﬁnal node stations. The process of cargo transportation is followed by the set rule of control. For such a model, one must determine possible modes of cargo transportation and describe their properties. This model is described by a ﬁnite-dimensional system of diﬀerential equations with nonlocal linear restrictions. The class of the solution satisfying nonlocal linear restrictions is extremely narrow. It results in the need for the “correct” extension of solutions of a system of diﬀerential equations to a class of quasi-solutions having the distinctive feature of gaps in a countable number of points. It was possible numerically using the Runge–Kutta method of the fourth order to build these quasi-solutions and determine their rate of growth. Let us note that in the technical plan the main complexity consisted in obtaining quasi-solutions satisfying the nonlocal linear restrictions. Furthermore, we investigated the dependence of quasi-solutions and, in particular, sizes of gaps (jumps) of solutions on a number of parameters of the model characterizing a rule of control, technologies for transportation of cargo and intensity of giving of cargo on a node station.

This proceedings publication is a compilation of selected contributions from the “Third International Conference on the Dynamics of Information Systems” which took place at the University of Florida, Gainesville, February 16–18, 2011. The purpose of this conference was to bring together scientists and engineers from industry, government, and academia in order to exchange new discoveries and results in a broad range of topics relevant to the theory and practice of dynamics of information systems. Dynamics of Information Systems: Mathematical Foundation presents state-of-the art research and is intended for graduate students and researchers interested in some of the most recent discoveries in information theory and dynamical systems. Scientists in other disciplines may also benefit from the applications of new developments to their own area of study.