Optimal Control of Volterra Difference Equations of the First Kind

We consider optimal control for Volterra Difference Equations of the form\begin{equation*}x(n + 1) = \sum\limits_{i = 0}^n B (i)x(n - i) + Cu(n),\quad n \in {\mathbb{Z}^ + }.\tag{1}\end{equation*}We show that the optimal control problem can be solved via a Riccati equation or alternatively, and computationally less involved, by solving a linear equation. We consider an application from epidemiology where the optimal control problem admits an optimal solution using the theoretical result. However, the optimal control does not necessarily satisfy constraints of the system?s biology, i.e. non-negativity of the state.