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On the continuity of the adjoint of evolution operators

Abstract By associating a quasi semigroup to a commutative evolution operator and using techniques from vector measures like the Gel’fand integral and the Radon- Nikodym theorem for Bochner integral, we prove the continuity on [0, +∞) of the adjoint of a certain class of evolution operators. Furthermore, we provide examples of evolution operators satisfying our conclusion. We also provide an application to optimal control theory; particularly, to an optimal control problem governed by a time-varying heat equation.