A bang-bang theorem with a finite number of switchings for nonlinear smooth control systems
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.
scientific article; zbMATH DE number 1273633