Hamilton–Pfaff type PDEs through multi-dimensional fractional optimization problems

In this paper, we derive Hamilton–Pfaff-type partial differential equations (PDEs) by employing exterior differential techniques within the framework of a multi-dimensional fractional optimal control problem, incorporating principles from fractional calculus. This approach provides a rigorous analytical foundation for studying systems governed by non-integer order dynamics, thereby enhancing the understanding of complex control processes. By formulating a control Hamiltonian 1-form associated with the underlying fractional optimal control problem and its corresponding adjoint distributions, we systematically derive the Hamilton–Pfaff-type PDEs. These equations capture the intricate relationship between fractional dynamics and optimal control, paving the way for further investigation of sophisticated systems in a wide range of applications.