In this paper the controllable system whose behaviour is described by a nonlinear Volterra integral equation, is studied. The set of admissible control functions is the closed ball of the space Lp (p > 1) with radius µ0 and centered at the origin. It is shown that the set of trajectories of the system is a bounded and precompact subset of the space of continuous functions.

Abstract. In this paper the control system described by a Urysohn type integral equation is studied. It is assumed that the control functions have integral constraint. Approximation of the set of trajectories generated by all admissible control functions is considered.Step by step way, the set of admissible control functions is replaced by a set consisting of a finite number of control functions which generates a finite number of trajectories. An evaluation of the Hausdorff distance between the set of trajectories of the system and the set, consisting of a finite number of trajectories is obtained.

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In this paper, the set of trajectories of the control system described by a nonlinear Volterra integral equation is studied. It is assumed that the set of admissible control functions is the closed ball of the space Lp, p > 1, with radius µ and centered at the origin. It is shown that the sections of the set of trajectories can be approximated by the sections of trajectories, generated by the mixed constrained and Lipschitz continuous control functions, the Lipschitz constant of which is bounded.