The properties and approximation of the set of trajectories of the control system described by nonlinear volterra integral equation

In this dissertation the control system described by a nonlinear Volterra integral equation is studied. It is assumed that the system is nonlinear with respect to the state and control vectors. The closed ball centered at the origin with radius $\mu$ in the space $L_p,$ $p>1,$ is chosen as the set of admissible control functions set. The properties of the set of trajectories of the control system and approximate construction of its sections are investigated. It is shown that the set of trajectories is a precompact subset of the space of continuous functions and it is proved that the set of trajectories depends on $\mu$ and $p$ continuously. The set of admissible control functions is replaced by a new compact control functions set which consists of integral constrained, geometric constrained and Lipschitz continuous control functions, the Lipschitz constant of which is bounded. It is shown that the set of trajectories generated by these control functions is a compact set. It is proved that the sections of the set of trajectories can be approximated by the sections of the set of trajectories, generated by the mixed constrained and Lipschitz continuous control functions, the Lipschitz constant of which is bounded.