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DCDS-B

This paper concerns an optimal control problem governed by a
linear evolution system with a small perturbation in the system
conductivity. The system without any perturbation is assumed to have
such a periodic property that it holds a periodic solution. In
general, the perturbed system dose not enjoy this periodic
property again, even though the perturbation has a small norm. The
goal of this research is to restore the periodic property for the
system, with a small perturbation, through utilizing such a control
that is optimal in certain sense. It also aims to study
characteristics of such an optimal control. The existence and
uniqueness of the optimal control is obtained. Furthermore, a
necessary and sufficient condition for the optimal control is
established.

DCDS

This paper is devoted to study the abstract functional differential equation (FDE) of the following form
$$\dot{u}(t)=Au(t)+\Phi u_t,$$
where $A$ generates a $C$-regularized semigroup, which is the generalization of $C_0$-semigroup and can be applied to deal with many important differential operators that the $C_0$-semigroup can not be used to.
We first show that
the $C$-well-posedness of a FDE is equivalent to the
$\mathscr{C}$-well-posedness of an abstract Cauchy problem in a product
Banach space, where the operator $\mathscr{C}$ is related with the
operator $C$ and will be defined in the following text. Then, by making use of a perturbation result of $C$-regularized semigroup, a sufficient condition is provided for the $C$-well-posedness of FDEs. Moreover, an
illustrative application to partial differential equation (PDE) with delay is given in the last section.

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