Dinkelbach Approach to Solving a Class of Fractional Optimal Control Problems

We consider optimal control problems with functional given by the ratio of two integrals (fractional optimal control problems). In particular, we focus on a special case with affine integrands and linear dynamics with respect to state and control. Since the standard optimal control theory cannot be used directly to solve a problem of this kind, we apply Dinkelbach’s approach to linearize it. Indeed, the fractional optimal control problem can be transformed into an equivalent monoparametric family {P_q} of linear optimal control problems. The special structure of the class of problems considered allows solving the fractional problem either explicitly or requiring straightforward classical numerical techniques to solve a single equation. An application to advertising efficiency maximization is presented. This work was partially supported by the Università Ca’ Foscari, Venezia, Italy, the MIUR (PRIN cofinancing 2005), the Council for Grants (under RF President) and State Aid to Fundamental Science Schools (Grant NSh-4113.2008.6). We thank Angelo Miele, Panos Pardalos and the anonymous referees for comments and suggestions.