Symmetry, Conservation Law, Uniqueness and Stability of Optimal Control and Inverse Problems for Burgers’ Equation

This paper tackles the ill-posed inversion of initial conditions and diffusion coefficient for Burgers’ equation with a source term. Using optimal control theory combined with a finite difference discretization scheme and a dual-functional descent method (DFDM), it sets the unknown boundary function g(τ) and diffusion coefficient u as control variables to build a multi-objective functional, proving the existence of the optimal solution via the variational method. Symmetry analysis reveals the intrinsic connection between the equation’s Lie group invariances and conservation laws through Noether’s theorem, providing a natural regularization framework for the inverse problem. Uniqueness and stability are demonstrated by the adjoint equation under cost function convexity. An energy-consistent discrete scheme is created to verify the energy conservation law while preserving the underlying symmetry structure. A comprehensive error analysis reveals dual error sources in inverse problems. A multi-scale adaptive inversion algorithm incorporating symmetry considerations achieves high-precision recovery under noise: boundary error <1%, energy conservation error ≈0.13%. The symmetry-aware approach enhances algorithmic robustness and maintains physical consistency, with the solution showing linear robustness to noise perturbations.