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On Bellman spheres for linear controlled objects of second order. (English) Zbl 0917.49002
Let be given the dynamic controlled object defined by $\dot{x}= Ax+Bu,\quad x \in \mathbb{R}^{n},\quad u \in U \subset R^{m},\tag{*}$ where the constant matrices define linear mappings $$A: \mathbb{R}^{n} \rightarrow \mathbb{R}^{n}$$, $$B: \mathbb{R}^{m} \rightarrow \mathbb{R}^{n}$$, the set of admissible controls $$U$$ is compact, convex and contains the origin in its interior. The measurable function $$u(t)$$, $$t \in [t_{0}, t_{1}]$$, is an admissible control if $$u(t) \in U$$ for each $$t \in t[t_{0}, t_{1}]$$. The time-optimal problem is to find an admissible control function $$u(t)$$ that brings the system (*) from any initial state $$x_{0}$$ to the origin in a minimum time $$t^{*} < \infty$$. Using the well-known ways of optimal problem solution (Bellman, Pontryagin) the second order linear object controlled by non-standard two-dimensional control function has been investigated. Some examples complete the received results.
Reviewer: W.Hejmo (Kraków)
##### MSC:
 49J15 Existence theories for optimal control problems involving ordinary differential equations 49K15 Optimality conditions for problems involving ordinary differential equations 49N05 Linear optimal control problems
##### Keywords:
time-optimal problem; maximum principle; Bellman’s sphere
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