A bang-bang theorem with a finite number of switchings for nonlinear smooth control systems
type: Item
instance of: scholarly article
title: A bang-bang theorem with a finite number of switchings for nonlinear smooth control systems
zbMATH Open document ID: 0926.49013
DOI: 10.1007/BF02355111
author: S. A. Vakhrameev
published in: Journal of Mathematical Sciences (New York)
publication date: 1999-04-07T00:00:00Z
review text: Let us consider the following classical result [cf. \textit{L. S. Pontryagin, V. G. Boltyanskij, R. V. Gamkrelidze} and \textit{E. F. Mishchenko}, ``The mathematical theory of optimal processes'', 4th ed. (Russian) (1983; Zbl 0516.49001; 1st ed. 1962; Zbl 0102.31901; English translation 1962; Zbl 0102.32001)] of the linear control theory (Gamkrelidze's theorem on the finiteness of the number of switchings), which is also known as the strong bang-bang theorem [cf. \textit{V. Veliov}, C. R. Acad. Bulg. Sci. 40, No. 2, 31-33 (1987; Zbl 0641.49015)]: The following linear control system \[ \dot{x} \;= \;Ax \;+ \;Bu, \;\;x\in R^n, \;u \in U, \] is considered, where \(U\) is a compact convex polyhedron in \(R^m\), and \(A\) and \(B\) are constant \(n \times n\) and \(n \times m\) matrices, respectively. The set of admissible controls is a suitable subset of the set of all measurable functions with values in \(U\). It is also assumed that the control system satisfies the so-called general position with respect to the polyhedron \(U\). Then every extremal control (i.e., every admissible control satisfying the Pontryagin maximum principle) coincides almost everywhere with a piecewise-constant function whose values are only the vertices of the polyhedron \(U\) (this result holds true without the assumption for general position with respect to \(U\) [cf. \textit{V. Veliov} (loc. cit.) for details]. In this paper, a generalization of this result for the nonlinear case is proved. Moreover, the connection of the obtained result with the results of other authors is discussed and some open questions are stated.
Mathematics Subject Classification ID: 49K15
Mathematics Subject Classification ID: 49K30
zbMATH DE Number: 1273633
zbMATH Keywords: smooth control system
zbMATH Keywords: piece-wise constant extremal control
zbMATH Keywords: bang-bang theorem
zbMATH Keywords: Pontryagin maximum principle
reviewed by: Mikhail Krastanov
MaRDI profile type: Publication
cites work: Q4195269
cites work: Q4871472
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cites work: A theorem on the finiteness of the switching number for smooth non-linear controlled systems
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cites work: Local invariants of smooth control systems
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cites work: A Generalized Estimate of the Number of Zeros for Solutions of a Class of Linear Differential Equations
cites work: A Bang-Bang Theorem with Bounds on the Number of Switchings
cites work: Q3782318
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description: scientific article; zbMATH DE number 1273633