Optimal control of the Cattaneo–Hristov heat diffusion model

In this study, the optimal control problem for Cattaneo–Hristov heat diffusion, a partial differential equation including both fractional-order Caputo–Fabrizio and integer-order derivatives, is formulated for a rigid heat conductor with finite length. The state and control functions denoting temperature and heat source, respectively, are represented by eigenfunctions to eliminate the spatial coordinate. The necessary optimality conditions corresponding to a time-dependent dynamical system are derived via Hamilton’s principle. Because the optimality system contains both integer-order and fractional-order left- and right-side Caputo–Fabrizio derivatives, it cannot be solved analytically. Therefore, a numerical method based on the Volterra integral approach combined with the forward–backward finite difference schemes is applied to solve the system. Finally, the physical behaviours of the temperature state and heat control under the variation of the fractional parameter are depicted graphically.