Causal state feedback representation for linear quadratic optimal control problems of singular Volterra integral equations

This paper is concerned with a linear quadratic optimal control for a class of singular Volterra integral equations. Our framework covers the problems for fractional differential equations. Under some necessary convexity conditions, an optimal control exists, and can be characterized via Fréchet derivative of the quadratic functional in a Hilbert space or via maximum principle type necessary conditions. However, these (equivalent) characterizations are not causal, meaning that the current value of the optimal control depends on the future values of the optimal state. Practically, this is not feasible. We obtain a causal state feedback representation of the optimal control via a Fredholm integral equation. Finally, a concrete form of our results for fractional differential equations is presented.