A Mesh-Independent Adjoint Consistency Defect in Optimal Control of the Caputo Time-Fractional Lindblad Equation: Sharp Classical-Limit Rate, Correction, and Convergence

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Time-fractional generalizations of the Lindblad master equation describe open quantum systems whose coupling to the environment exhibits power-law memory. We develop the optimal-control theory of such systems and analyse the consistency of the adjoint calculus on which every gradient-based pulse-design method relies. Casting the density operator in a fractional Bochner–Sobolev space of Hilbert–Schmidt operator valued functions, we establish well-posedness through Mittag–Leffler resolvent families, prove that the completely positive trace-preserving (CPTP) structure is preserved for the controlled, time-dependent generator without recourse to subordination, and obtain existence and uniqueness of optimal controls by the direct method. The adjoint is governed by the right Riemann–Liouville derivative with a fractional-integral terminal condition, a structure established for Caputo dynamics with a Mayer cost by Bergounioux and Bourdin, who also showed that a pointwise terminal costate cannot exist. Our central result concerns the discrete counterpart of that fact, where existence is never lost: imposed on the right-Caputo adjoint of a convergent scheme, the pointwise condition yields a bounded costate and a well-defined reduced gradient carrying an error that is mesh-independent. We further establish a sharp rate in the classical limit: the defect vanishes exactly linearly, ∥Δ(β)∥=C(1−β)+O((1−β)2), with C given in closed form through a digamma series. Two consequences follow: monotonicity of the defect in the memory order is proved near β=1, and the memory order is locally identifiable from gradient data alone. A corrected adjoint restores consistency with proven convergence rates. Numerical experiments on two-level, three-level and two-qubit open systems (Liouville dimension up to 16) confirm the mesh-independence, reproduce C to three significant digits, and recover the full rate on graded meshes.