  In this paper, we study the optimal control problem with terminal and
inequality state constraints for state equations described by Volterra integral
equations having singular and nonsingular kernels. The singular kernel
introduces abnormal behavior of the state trajectory with respect to the
parameter of $\alpha \in (0,1)$. Our state equation is able to cover various
state dynamics such as any types of Volterra integral equations with
nonsingular kernels only, fractional differential equations (in the sense of
Riemann-Liouville or Caputo), and ordinary differential state equations. We
obtain the well-posedness (in $L^p$ and $C$ spaces) and precise estimates of
the state equation using the generalized Gronwall's inequality and the proper
regularities of integrals having singular and nonsingular integrands. We then
prove the maximum principle for the corresponding state-constrained optimal
control problem. In the derivation of the maximum principle, due the presence
of the state constraints and the control space being only a separable metric
space, we have to employ the Ekeland variational principle and the spike
variation technique, together with the intrinsic properties of distance
functions and the generalized Gronwall's inequality, to obtain the desired
necessary conditions for optimality. In fact, as the state equation has both
singular and nonsingular kernels, the maximum principle of this paper is new,
where its proof is more involved than that for the problems of Volterra
integral equations studied in the existing literature. Examples are provided to
illustrate the theoretical results of this paper.
