A Necessary Optimality Condition for Optimal Control of Caputo Fractional Evolution Equations

Abstract

In this paper, we prove the Pontryagin maximum principle, which constitutes the necessary optimality condition, for the infinite-dimensional optimal control problem of X-valued left Caputo fractional evolution equations, where X is a Banach space. An important step in the proof to obtain the desired Hamiltonian maximization condition is to establish new variational and duality analysis. While the former is characterized by a linear X-valued left Caputo fractional evolution equation via spike variation, the latter requires the adjoint equation characterized by a linear X∗-valued right Riemann-Liouville (RL) fractional evolution equation, where X∗ is a dual space of X. We show the variational and duality analysis with the help of the infinite-dimensional fractional version of the technical lemma and the explicit representation of solutions to linear (Caputo and RL) fractional evolution equations using left and right RL state-transition evolution operators.