Multi-order fractional optimal control systems: existence and optimality conditions

We study an optimal control problem for dynamical systems with multiorder (multiindex) fractional dynamics. In this setting, different components of the state vector evolve with different fractional orders, meaning that each state variable is governed by its own memory intensity. This allows the model to describe systems where different physical quantities exhibit different rates of memory and hereditary effects. Such a framework is motivated by applications in which a single uniform memory law is not realistic. Examples include heterogeneous diffusion processes, viscoelastic materials with multiple relaxation behaviors, and coupled dynamical systems with component-dependent memory effects. These situations cannot be adequately captured by classical integer-order or standard single-order fractional models. Using Caputo fractional derivatives, we formulate a general class of multi-order fractional optimal control systems with quadratic cost functionals. The main contributions of this work include the derivation of necessary optimality conditions via a fractional Pontryagin Maximum Principle adapted to the multiorder setting, and the resulting coupled state-adjoint system with heterogeneous fractional dynamics. In addition, we obtain an explicit optimal control characterization in the linear-quadratic case, extending classical fractional optimal control results to the multiorder framework. The proposed results generalize existing single-order fractional control theory and provide a more flexible mathematical tool for modeling and analyzing systems with multiple interacting memory effects. This may open new directions for both analytical and numerical studies of complex fractional dynamical systems.