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**Convexity of the set of fixed points generated by some control systems.**
*(English)*
Zbl 1175.93102

Summary: We deal with an application of the fixed point theorem for nonexpansive mappings to a class of control systems. We study closed-loop and open-loop controllable dynamical systems governed by Ordinary Differential Equations (ODEs) and establish convexity of the set of trajectories. Solutions to the above ODEs are considered as fixed points of the associated system-operator. If convexity of the set of trajectories is established, this can be used to estimate and approximate the reachable set of dynamical systems under consideration. The estimations/approximations of the above type are important in various engineering applications as, for example, the verification of safety properties.

### MSC:

93C15 | Control/observation systems governed by ordinary differential equations |

47H10 | Fixed-point theorems |

54H25 | Fixed-point and coincidence theorems (topological aspects) |

### Keywords:

fixed point theorem for nonexpansive mappings; control systems; ordinary differential equations; convexity of the set of trajectories
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\textit{V. Azhmyakov}, J. Appl. Math. 2009, Article ID 291849, 14 p. (2009; Zbl 1175.93102)

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