On the existence of admissible processes for control systems with mixed constraints

A control system with mixed equality-type constraints and end-point constraints is considered. In terms of the generalized Jacobian (Clarke's derivative) with respect to the control variable of the mapping defining the constraints, sufficient conditions for the existence of continuous admissible positional controls are obtained. The proof of the corresponding theorem is based on reducing the control system to a boundary value problem for an ordinary differential equation via a nonlocal implicit function theorem. This problem is then reduced to the problem of finding a fixed point of a continuous mapping defined on a finite-dimensional closed ball and to applying an analogue of Brouwer's fixed point theorem. In addition, a control system with mixed inequality-type constraints and end-point constraints is studied. In terms of the first derivatives with respect to the control variable of the functions that define the constraints, sufficient conditions for the existence of continuous admissible positional controls are also obtained. The proof of the corresponding theorem is carried out by passing from a system of smooth inequality-type constraints to one locally Lipschitz equality-type constraint.