Optimal Control of the Inverse Problem of the Fractional Burgers Equation

This paper investigates the well-posedness of the inverse problem for the time-fractional Burgers equation, which aims to reconstruct initial conditions from terminal observations. Such equations are crucial for the modeling of hydrodynamic phenomena with memory effects. The inverse problem involves inferring initial conditions from terminal observation data, and such problems are typically ill-posed. A framework based on optimal control theory is proposed, addressing the ill-posedness via H1 regularization. Three substantial results are achieved: (1) a rigorous mathematical framework transforming the ill-posed inverse problem into a well-posed optimization problem with proven existence of solutions; (2) theoretical guarantee of solution uniqueness when the regularization parameter is α>0 and the stability is of order O(δ) with respect to observation noise (δ); and (3) the discovery of a “super-stability” phenomenon in numerical experiments, where the actual stability index (0.046) significantly outperforms theoretical expectations (1.0). Finally, the theoretical framework is validated through comprehensive numerical experiments, demonstrating the accuracy and practical effectiveness of the proposed optimal control approach for the reconstruction of hydrodynamic initial conditions.