In this work we consider measure-valued optimal control problems involving the KdV-Burgers (KdVB) equation as a state equation. These optimal control problems are motivated by an inverse problem and a control problem involving the flow of water in a channel over topography. Well-posedness of the optimal control problem is established which involves the investigation of the KdVB equation for irregular source terms. Moreover optimality conditions for the control problem are derived. Efficient numerical schemes based on spectral methods are proposed for the state and adjoint equation, as well as adequate optimization methods. The theoretical results are illustrated by two numerical examples.