On Diferential Inclusions Arising from Some Discontinuous Systems
type: Item
instance of: scholarly article
title: On Diferential Inclusions Arising from Some Discontinuous Systems
DOI: 10.1080/01630563.2024.2333251
author: Alexander Fominyh
published in: Numerical Functional Analysis and Optimization
publication date: 2024-08-26T00:00:00Z
review text: The author considers a control system of the following form\N\[\N\dot{x} = Ax + Bu\N\]\Non \([0,T]\) with the boundary conditions\N\[\Nx(0) = x_0,\quad x_j(T) = x_{T_j},\N\]\Nwhere \(j\) belongs to a given index set \(J \subset \{1,\dots,n\}\). Here, \(A\) is a constant \(n \times n\) matrix, \(B\) has the form \(B = \mathrm{diag}[E_m,\mathbb{O}_{n-m}],\) \(m < n\). It is supposed also that the discontinuity surface \(s(x,c) = 0_m\) is given, where the parameters \(c \in \mathbb{R}^l\) are unknown.\N\NUnder these conditions, it is assumed that the control functions have the form\N\[\Nu_i = - \alpha_i |x| \mathrm{sign} (s_i(x,c)), \quad \alpha_i \in [\underline{a}_i,\overline{a}_i],\,\,i = 1,\dots,m.\N\]\NThen on the surfaces \(s_i(c,x) = 0,\) \(i = 1,\dots,m\), the system takes the form of the differential inclusion \N\[\N\begin{cases} \dot{x}_i \in A_ix + [- \overline{a}_i,\overline{a}_i]|x|,\quad i = 1,\dots,m; \\\N\dot{x}_i \in A_ix, \quad i = m+1,\dots,n. \end{cases} \N\]\NIt is required to find such a trajectory which moves along the discontinuity surface (the parameters \(c\) are to be determined as well) and satisfies the above differential inclusion and boundary conditions.\N\NThis task is reduced to a variational problem of finding a minimum of certain functionals defined on functional spaces. Some differentiability properties of these functionals are considered and necessary conditions of a minimum are presented.
reviewed by: Valerii V. Obukhovskij
Mathematics Subject Classification ID: 34A60
Mathematics Subject Classification ID: 34H05
Mathematics Subject Classification ID: 34A36
Mathematics Subject Classification ID: 47J30
zbMATH DE Number: 7901797
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 7901797