Problems of control of distributions in dynamical systems
type: Item
instance of: scholarly article
title: Problems of control of distributions in dynamical systems
zbMATH Open document ID: 0946.93044
author: A. V. Pukhlikov
published in: Automation and Remote Control
publication date: 1999-09-15T00:00:00Z
review text: Some problems of optimal control of distributions are considered. One of the most characteristic problem of this type may be formulated as follows: The dynamic of the system is given by the equation \[ \dot x(t)= f(x(t), u(t))\quad\text{with }x(t)\in \mathbb{R}^n\quad\text{and }u(t)\in U\subset \mathbb{R}^n.\tag{1} \] The performance index is of the form \[ \int_{\mathbb{R}^n} \phi(x)\zeta(T, x) dx\to \min,\tag{2} \] where the function \(\zeta= \zeta(t, x)\), \(t\in [0,T]\), \(x\in\mathbb{R}^n\) satisfies the continuity equation \[ {\partial\over\partial t} \zeta(t,x)= \text{div}_x[\zeta(t, x) f(x,u)]= 0\tag{3} \] and \[ \int_{\mathbb{R}^n} \zeta(t, x) dx\equiv 1,\quad \zeta(x,t)\geq 0. \] The function \(\zeta(0,y)= \zeta_0(y)\) is interpreted as the probability density that the initial value for the dynamical system (1) takes the value \(x(0)= y\) at the moment \(t= 0\). For systems of type (1)--(3) some necessary optimality conditions in the form of the Pontryagin maximum principle and some analogs of the Feldbaum theorem on the number of switchings are proved.
Mathematics Subject Classification ID: 93E20
Mathematics Subject Classification ID: 49K15
Mathematics Subject Classification ID: 49K20
Mathematics Subject Classification ID: 93C20
zbMATH DE Number: 1290433
zbMATH Keywords: optimal control of distributions
zbMATH Keywords: probability density
zbMATH Keywords: maximum principle
zbMATH Keywords: Feldbaum theorem
zbMATH Keywords: switchings
reviewed by: Stanislaw Walczak
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 1290433