On the Reconstruction of a Parameter of an Elliptic System

The problem considered consists of the reconstruction (restoration) of a parameter of an elliptic system based on the results of measuring its state. To solve this problem the method of dynamical approximation is used. The method was proposed by A.V. Kryazhimskii and the author, and is based on the ideas of the differential games theory and the ill-posed problems theory. The reconstruction algorithm presented here is stable with respect to the errors of measurements and is fairly constructive

On dynamical regularization under random noise

We consider the problem of constructing a robust dynamic approximation of a time-varying input to a control system from the results of inaccurate observation of the states of the system. In contrast to the earlier studied cases in which the observation errors are assumed to be small in the metric sense, the errors in the present case are allowed to take, generally, large values and are subject to a certain probability distribution. The observation errors occurring at different instants are supposed to be statistically independent. Under the assumption that the expected values of the observation errors are small, we construct a dynamical algorithm for approximating the normal (minimal in the sense of the mean-square norm) input; the algorithm ensures an arbitrarily high level of the mean-square approximation accuracy with an arbitrarily high probability

Input Reconstructibility for Linear Dynamics. Ordinary Differential Equations

The paper deals with the standard input-output observation scheme for a dynamic system governed by a linear ordinary differential equation. The initial problem is to reconstruct the actually working time-varying input, given a state observation result. Normally, the problem has no solution: observation is too poor to select the real input from the collection of "possible" ones. It is proposed to turn the problem as follows: what information of the real input is reconstructible precisely? The dual setting: what information of the real input is totally non-reconstructible? The question of aftereffect arises naturally: does accumulation of observation results lead to the informational jump -- from nonreconstructibility to complete reconstructibility -- in the past? Posing and answering these questions is the goal of the present study

Inverse Problem of Dynamics for Systems Described by Parabolic Inequality

This paper deals with a specific inverse problem of dynamics for a system described by a parabolic inequality. The aim is to reconstruct the input (the control) of the system on the basis of an on-line measurement corrupted by an error. The techniques applied to the solution are a combination of those developed in positional control theory and the theory of ill-posed problems

On the Reconstruction of a Parameter for a Hyperbolic System

This paper addresses the problem of on-line identification of a parameter of a distributed hyperbolic system through available continuous measurements. The solution is achieved here by introducing an adjoint dynamic model with feedback control developed on the basis of the observation data. The suggested on-line reconstruction algorithm ensures numerical stability of the procedure and leads to effective simulation results

Regularized extremal shift in problems of stable control

We discuss a technical approach, based on the method of regularized extremal shift (RES), intended to help solve problems of stable control of uncertain dynamical systems. Our goal is to demonstrate the essence and abilities of the RES technique; for this purpose we construct feedback controller for approximate tracking a prescribed trajectory of an inaccurately observed system described by a parabolic equation. The controller is "resource-saving" in a sense that control resource spent for approximate tracking do not exceed those needed for tracking in an "ideal" situation where the current values of the input disturbance are fully observable. © 2013 IFIP International Federation for Information Processing.German Sci. Found. (DFG) Eur. Sci. Found. (ESF);Natl. Inst. Res. Comput. Sci. Control France (INRIA);DFG Research Center MATHEON;Weierstrass Institute for Applied Analysis and Stochastics (WIAS);European Patent Offic

Inverse Problems for Ordinary Differential Equations: Dynamical Solutions

This monograph provides an extensive account of the techniques used to solve a wide range of problems in the mathematics of dynamical systems operating under unpredictable time-varying disturbances. Starting from basic motivations and principles, the authors rapidly present more advanced models compatible with a study of dynamics and stability in nondegenerate and combined systems

On the solvability of problems of guaranteeing control for partially observable linear dynamical systems

This paper is devoted to a specification of the method of open-loop control packages, a universal instrument for verification of the solvability of problems of closed-loop control for partially observable dynamical systems. Under the assumption that the control system and observed signal are linear and the set of the admissible initial states is finite, a structure of the corresponding open-loop control packages is specified and a finite-step backward construction is described, which provides a criterion for the solvability of a problem of guaranteed closed-loop guidance onto a target set at a prescribed time