On the Controllability of Non-linear Systems* Sur l'aptitude au r~glage des syst~mes non-lin6aires
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H. TOKUMARUt and N. ADACHI t
For a non-linear control system there exists a corresponding lower dimensional control system such that the controllabifity property of the system is equivalent to that of the original one.
Summary--Some discussions for the controllability in nonlinear control systems are presented. The control systems treated are described by ordinary differential equations. Several concepts concerning the controllability are introduced. If every initial state of the system can be transferred to the origin in a finite time, the system is called "controllable". If the time required is infinite, then the system is "quasi-controllable". If the system has the controllability property in the neighborhood of the stationary state of the system, then the system is "locally controllable". By the definitions, if the system is quasi-controllable and locally controllable, then, is controllable. Our discussions are restricted to the systems in which controls operate linearly. Such systems, either linear or non-linear are called control systems with controls appearing linearly.
Under suitable conditions the quasi-controllability of the given system of such type can be reduced to that of a certain lower dimensional control system. Hence, the controllability analysis can be very simplified.
Using this result, the controllability of some special types of non-linear systems are considered in detail and sufficient conditions for the quasi-controllability are obtained. In the last section, some examples are presented. For these examples, sufficient conditions for the controllability are obtained, connecting the conditions for the quasi-controllability and the local controllability.