In this manuscript, we examine a fractional inverse problem of order 0 < ρ < 1 {0<\rho<1} in a Banach space, including the Atangana–Baleanu fractional derivative in the Caputo sense. We use an overdetermined condition on a mild solution to identify the parameter. The major strategies for determining the outcome are a direct approach using the Volterra integral equation for sufficiently regular data. For less regular data, an optimal control approach uses Euler–Lagrange (EL) equations for the fractional order control problem (FOCP) and a numerical approach for solving FOCP. At last, a numerical example is provided in the support of our results.

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