Extremal Shift Method Research Articles

Approximate calculation of the Caputo-type fractional derivative from inaccurate data. Dynamical approach

Abstract A specific formulation of the “classical” problem of mathematical analysis is considered. This is the problem of calculating the derivative of a function. The purpose of this work is to construct an algorithm for the approximate calculation of the Caputo-type fractional derivative based on the methods of control theory. The input data of the algorithm is represented by inaccurate measured function values at discrete, frequently enough, times. The proposed algorithm is based on two aspects: a local modification of the Tikhonov regularization method from the theory of ill-posed problems and the Krasovskii extremal shift method from the guaranteed control theory, both of which ensure the stability to informational noises and computational errors. Numerical experiments were carried out to illustrate the operation of the algorithm.

Extremal Shift in a Problem of Tracking a Solution of an Operator Differential Equation

A control problem for an operator differential equation in a Hilbert space is considered. The problem consists in constructing an algorithm generating a feedback control and guaranteeing that the solution of the equation follows a solution of another equation, which is subject to an unknown disturbance. We assume that both equations are given on an infinite time interval and the unknown disturbance is an element of the space of square integrable functions; i.e., the perturbation may be unbounded. We construct two algorithms based on elements of the theory of ill-posed problems and the extremal shift method known in the theory of positional differential games. The algorithms are stable with respect to information noises and calculation errors. The first and second algorithms can be used in the cases of continuous and discrete measurement of solutions, respectively.

Tracking the Solution of a Linear Parabolic Equation Using Feedback Laws

We consider the problem of tracking the solution of a parabolic equation with an unknown right-hand side by the solution of a similar parabolic equation. To solve this problem, we propose two noise-resistant algorithms based on the extremal shift method known in guaranteed control theory. The first algorithmpertains to the case of continuousmeasurement of solutions to the equations, and the second, to the case of discrete measurement.

Dynamic Right-Hand Side Reconstruction Problem for a System of Fractional Differential Equations

For a controlled dynamical system described by nonlinear fractional differential equations, we consider the problem of dynamic reconstruction of an unknown disturbance from approximate measurements of the trajectory. For this problem, we construct an algorithm based on Krasovskii's extremal shift method and find its convergence rate.

On a Control Problem for a Linear System with Measurements of a Part of Phase Coordinates

We consider a control problem for a system of linear ordinary differential equations. It is required to design a feedback control procedure under which the rate of change of a part of the phase coordinates of the system would track the rate of change of a part of the phase coordinates of another system subject to an unknown disturbance. It is assumed that a part of phase coordinates of each of the systems is measured with error at discrete times. We propose a solution algorithm that is stable with respect to informational noises and computation errors. The algorithm is based on the extremal shift method known in the theory of guaranteed control. Since it is impossible to apply the “classical” extremal shift due to the incompleteness of the information on the phase coordinates, we propose a modification of this method that employs elements of the dynamic inversion theory. The latter is based on constructions from the theory of ill-posed problems. In the concluding section of the paper, we specify a class of systems nonlinear in the phase coordinates for which the algorithm is applicable.

On an algorithm for the problem of tracking a trajectory of a parabolic equation

Abstract In this paper, we consider the problem of tracking a solution of a reference parabolic equation by a solution of another equation. A stable algorithm based on the extremal shift method is proposed for this problem. The algorithm is designed to work on a sufficiently large time interval where both equations operate.

On a guaranteed guidance problem under incomplete information

We discuss the problem of guaranteed guidance of a linear control system by a fixed time under the assumption that the system is subject to an unknown disturbance. We consider the case when a part of state coordinates are measured and the set of unknown initial states is finite. We specify a solution algorithm based on the combination of the package approach, the theory of dynamic inversion, and the extremal shift method.

On a modification of the extremal shift method for a second-order differential equation in a Hilbert space

A problem of tracking a solution of a second-order differential equation in a Hilbert space by a solution of another equation is considered. It is assumed that the first (reference) equation is subject to the action of an unknown control, which is unbounded in time. In the case when the current states of both equations are observed with small errors, a solution algorithm stable with respect to informational noises and computational inaccuracies is designed. The algorithm is based on N.N. Krasovskii’s extremal shift method known in the theory of guaranteed control.

Infinite-horizon boundary control of distributed systems

For a boundary controlled dynamic system, algorithms for solving the problem of tracking reference motion and the problem of tracking reference control are described. The algorithms are robust to information noise and computational errors. The solution method is based on the extremal shift method from the theory of positional differential games.

An extremal shift method for control of HIV infection dynamics

AbstractOptimal control problems for mathematical models describing HIV infection dynamics in a human body are considered in the paper. An overview of current approaches to solution of control problems for models of HIV dynamics is presented for techniques related to construction of optimal programme (open loop) or positional (feedback) controls for various criteria of control process quality and is based on the Pontryagin’s maximum principle and the Bellman’s theory of dynamic programming, respectively. In the framework of the theory of positional differential games of Krasovskii, there are constructive and efficient methods of synthesis of controls for different classes of dynamical systems. In this paper we present a formalization of a control problem for HIV infection considered as a corresponding positional differential game. The control algorithm based on the method of extremal shift is applied to the ODEs model of HIV infection. The numerical implementation of the extremal shift method is developed to construct the control taking the system to a neighbourhood of a given trajectory using the information on the dynamics of the guides or leaders.

On designing a reconstruction-control algorithm for an ecological-economic model

A control problem for the Nordhaus ecological-economic model is considered. The problem consists in constructing an algorithm for the stable generation of control characteristics that provide a given quality of the process in the case when only a part of the phase coordinates are measured and uncontrolled disturbances act on the system. We design such an algorithm (a reconstruction-control controller) using the feedback control method. The algorithm consists of two blocks, the online reconstruction and positional control, which are based on appropriate modifications of N.N. Krasovskii’s extremal shift method.

Numerical Method for a Phase Constraint Optimization Problem in Macroeconomics

The article considers optimization of mixed constraints for a controlled system that describes countrylevel economic growth driven by technological progress. An algorithm is proposed, reducing this problem to an equivalent general optimization problem in a normed space, which in turn is solved by an iterative method based on N. N. Krasovskii's extremal shift method. Calculation results are reported for the economy of Japan in the second half of the twentieth century.

The tracking of the trajectory of a dynamical system

The problem of tracking the trajectory of a dynamical system, described by a vector differential equation, is considered. An algorithm for solving this problem, based on the Krasovskii extremal shift method, well-known in position control theory, is proposed.

Tracking a reference solution of a control system of phase field equations

We consider the problem of tracking a reference solution of a dynamical system described by a pair of distributed differential equations, the phase field equations. To solve this problem, we propose an algorithm based on Yu.S. Osipov’s theory of dynamic inversion and on N.N. Krasovskii’s extremal shift method developed in the theory of positional differential games.

On the reconstruction of unknown characteristics of a distributed system using a regularized extremal shift

The problem of a stable dynamical reconstruction of unknown characteristics in a distributed system described by a pair of differential equations is considered. The aim of the paper is to construct a solution algorithm for this problem. The proposed algorithm is based on Yu.S. Osipov’s theory of dynamic inversion and N.N. Krasovskii’s extremal shift method.

N.N. Krasovskii’s extremal shift method and problems of boundary control

For the boundary-controlled dynamic system obeying a parabolic differential equation with the Neumann boundary condition, the problems of following the reference motion, following the reference control, and guaranteed control (at domination of the controller resource) were solved on the basis of the N.N. Krasovskii method of extremal shift from the theory of positional differential games.

Extremum Problems with Separable Graphs

The paper introduces a geometric feature of separability of graphs for extremum equality-type boundary problems. To find an optimal value for a problem with an almost separable graph, the paper presents an iteration algorithm, each step of which minimizes Lagrangian function for the main variable with a fixed Lagrangian multiplier. This algorithm dates back to Krasovskii extremal shift method from differential game theory.

Boundary Control Problems with Convex Cost Criterion

A class of boundary-distributed linear control systems in Banach spaces is studied. A maximum principle for a convex control problem associated with such systems is obtained.