Doubly Controlled System with Hysteresis and Delay

We consider a control system described by an ordinary differential equation and an ordinary differential inclusion including a hysteresis operator of generalized play type and a time delay in the perturbation terms. The equation and inclusion of the system are both controlled, but with controls of different types. The control in the inclusion is subject to a convex-valued upper semicontinuous control constraint, while the control in the equation, to a nonconvex-valued lower semicontinuous constraint. We prove the existence of a solution to the system.