  Inverse problems of recovering space-dependent parameters, e.g., initial
condition, space-dependent source or potential coefficient, in a subdiffusion
model from the terminal observation have been extensively studied in recent
years. However, all existing studies have assumed that the terminal time at
which one takes the observation is exactly known. In this work, we present
uniqueness and stability results for three canonical inverse problems, e.g.,
backward problem, inverse source and inverse potential problems, from the
terminal observation at an unknown time. The subdiffusive nature of the problem
indicates that one can simultaneously determine the terminal time and
space-dependent parameter. The analysis is based on explicit solution
representations, asymptotic behavior of the Mittag-Leffler function, and mild
regularity conditions on the problem data. Further, we present several one- and
two-dimensional numerical experiments to illustrate the feasibility of the
approach.
