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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
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