In this study, we presented a novel fractional nonlocal thermoelastic heat conduction model that extends the Guyer–Krumhansl framework by incorporating size-dependent nonlocal thermal effects and non-Fourier heat conduction characteristics. The model extends the traditional approach using the single-phase-lag (SPL) method derived from Moore–Gibson–Thompson (MGT) heat theory. By employing the Atangana–Baleanu (AB) fractional derivative with a non-singular kernel, we integrated nonlocal features through fractional derivatives, enhancing its applicability to complex thermal behaviors in materials exhibiting combined nonlocal and fractional dynamics. To validate the model, thermoelastic interactions were examined in a long, hollow cylinder subjected to a uniform electromagnetic field. The outer surface was thermally insulated and traction-free, while the inner surface, also traction-free, experienced thermal shock. Governing equations were solved using the Laplace transform method, and numerical solutions were obtained via the Dubner–Abate algorithm. The results were compared with conventional and generalized thermoelastic models to assess accuracy and effectiveness. Additional analysis explored material properties through graphical data, considering various fractional orders and operators, thereby enriching the understanding of system behavior under different conditions. The findings demonstrated the advantages of the fractional nonlocal thermoelastic model in capturing complex thermal interactions within advanced materials, contributing significantly to heat conduction theory.

This work was devoted to a minimum-energy optimal control problem governed by a fourth-order reaction–diffusion equation in which the control acts through a drift term. Such bilinear control problems arise naturally in transport phenomena and higher-order diffusion models, but their analysis is challenging because of the nonlinear coupling between the state and the control, as well as the difficulty of characterizing controls that are truly optimal with respect to energy. To overcome this difficulty, we formulated a family of penalized optimal control problems and characterized the minimum energy control as the limit of their solutions. The analysis was based on the associated optimality system, involving the state and adjoint equations, together with tools from convex optimization. Under appropriate assumptions, we established the existence and uniqueness of the optimal control, which provides a well-posed framework for the problem under consideration. For the numerical approximation, the resulting optimality system was solved by means of a conjugate gradient algorithm that requires, at each iteration, only the solution of the state equation and the adjoint equation. Numerical simulations showed a clear decrease in the tracking error and indicated that the computed state approaches the desired configuration with a small relative error after convergence. These results confirmed the stability, accuracy, and efficiency of the proposed approach for the minimum-energy control of fourth-order advection–reaction–diffusion systems.