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Bang-bang control of quasi-differential equations. (English) Zbl 0685.49008

Modern optimal control, Conf. in Honor of S. Lefschetz and J. P. LaSalle, Lect. Notes Pure Appl. Math. 119, 301-313 (1989).
[For the entire collection see Zbl 0673.00014.]
Consider the linear control system \(\dot x=A(t)x+B(t)u\) with state vector \(x\in {\mathbb{R}}^ n\), control vector \(u\in {\mathbb{R}}^ m\), at each time \(t\in {\mathcal I}\), an open interval in \({\mathbb{R}}\). The coefficient matrices A(t), B(t) are merely locally integrable, but have a particular format generalizing the companion form of a scalar equation, and permitting a description in the notation of quasi-differential control equations.
The system is proved to be fully controllable in \({\mathbb{R}}^ n\) (on arbitrarily short durations). If \(| u(t)| \leq 1\), then optimal controllers are proved to exist (for the minimal-time regulator problem), and are shown to have the bang-bang property.
Reviewer: L.Markus

MSC:

49J30 Existence of optimal solutions belonging to restricted classes (Lipschitz controls, bang-bang controls, etc.)
49J99 Existence theories in calculus of variations and optimal control

Citations:

Zbl 0673.00014