Abstract
Many results for nonlinear systems have been obtained concerning the existence of an optimal control under the assumptions that a certain convexity condition holds and the system is controllable. However, it is not at all obvious when a given nonlinear system is controllable, particularly, if the system operates only over a limited time interval.
In this paper, we first formulate this problem as an existence problem of fixed points for two nonlinear operator equations with some parameters in function spaces.
On some assumptions, we show first that each nonlinear operator has a fixed point; x=Pv(x), u=Qy(u), and second that there exists a set of fixed points which simultaneously satisfy the two nonlinear operator equations; x=Pu(x), u=Qx(u). Finally, it is shown that the pair of above mentioned fixed points is a solution of the original controllability problem, from which a sufficient condition is derived for controllability of nonlinear systems.
