Existence and uniqueness of solutions to rate independent systems with history variable

This paper investigates rate-independent systems (RIS), where the dissipation functional depends not only on the rate but also on the history of the state. The latter is expressed in terms of an integral operator. We establish an existence result for the original problem and for the control thereof, without resorting to smallness assumptions. Under a smoothness condition, we prove the uniqueness of solutions to a certain class of history-dependent RIS where the subdifferential of the dissipation potential is an unbounded operator. In this context, we derive an essential estimate that opens the door to future research on the topic of optimization.