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.

在这项研究中，Cattaneo-Hristov 热扩散的最优控制问题是一个偏微分方程，包括分数阶 Caputo-Fabrizio 和整数阶导数，用于有限长度的刚性热导体。分别表示温度和热源的状态和控制函数由特征函数表示以消除空间坐标。对应于瞬态动力系统的必要优化条件是通过哈密顿原理推导出来的。由于最优系统同时包含整数阶和分数阶的左右 Caputo-Fabrizio 导数，因此无法解析求解。因此，采用基于 Volterra 积分方法结合前向-后向有限差分格式的数值方法来求解该系统。