Compactness of the set of trajectories of the control system described by a Urysohn type integral equation
The control system with integral constraint on the controls is studied, where the behavior of the system by a Urysohn type integral equation is described. It is assumed that the system is nonlinear with respect to the state vector, affine with respect to the control vector. The closed ball of the space Lp(E; R m) (p > 1) with radius r and centered at the origin, is chosen as the set of admissible control functions, where E ⊂ R k is a compact set. It is proved that the set of trajectories generated by all admissible control functions is a compact subset of the space of continuous functions.
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