Operator theoretic approach to optimal control problems characterized by the Caputo fractional differential equations

Abstract

In this paper, our aim is to solve an optimal control problem by using an operator theoretic approach in the sense of Caputo fractional derivative. We first reduce the fractional dynamical system into an equivalent Hammerstein operator equation and then we provide a sufficient condition for the operators to ensure the existence of an optimal pair for the fractional dynamical system. By analyzing minimum conditions for the fractional optimal control problem (FOCP), the existence results of an optimal pair are obtained for both convex and non-convex cases. We derive an optimality system with a quadratic cost functional by using the Gateaux derivative. Finally, we relate our optimality system with the Hamiltonian system of minimum principle, and some examples are included to demonstrate the effectiveness of theoretical results. The primary motivation for this research is to investigate a new approach to solving the fractional optimal control problem employing functional analysis and operator theory techniques.

Keywords

• Fractional optimal control
• Gateaux derivative
• Optimality conditions
• Functional analysis
• Caputo fractional derivative
• Mittag-Leffler function