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Finite-dimensional open-loop control generators for nonlinear systems. (English) Zbl 0641.93035
This paper is concerned with nonlinear control systems $$\dot x=f(x,u)$$ with the state x belonging to an open part of $${\mathbb{R}}^ n$$ and the control u belonging to $${\mathbb{R}}^ m$$. The purpose of the paper is to discuss open-loop control generators. A control generator is an autonomous system of the form $${\dot \omega}=P(\omega)$$ yielding an output $$u=Q(\omega)$$. In many practical circumstances the controls of the original system are generated in this way. The main result of the paper is that under fairly general conditions one can assure that given some trajectory of $$\dot x=f(x,u)$$ connecting any two states $$x_ 0$$ and $$x_ 1$$ for some u then one can also find an open-loop control generator together with a properly chosen initial state $$\omega_ 0$$ which yields a trajectory connecting $$x_ 0$$ and $$x_ 1$$. The most important requirement on the system $$\dot x=f(x,u)$$ that this control generator exists is that the system is controllable.
Reviewer: H.Nijmeijer

##### MSC:
 93C10 Nonlinear systems in control theory 93B15 Realizations from input-output data 93B17 Transformations 93B05 Controllability 93C15 Control/observation systems governed by ordinary differential equations
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