Identification of diffusion coefficient in a semi-linear parabolic equation with incomplete initial condition: a no-regret control method

This paper addresses the problem of identifying the unknown diffusion coefficient in a semi-linear parabolic equation with incomplete initial condition. We propose an optimal control approach using the no-regret control method and the adapted low-regret control. Our approach provides a full characterisation of the unknown diffusion coefficient independent of the missing initial condition. We also present an optimality system that describes the adapted low-regret control and use it to find a full description of the no-regret control by taking the limit of the sequence of adapted low-regret controls.