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A hidden-memory variable-order time-fractional optimal control model: analysis and approximation. (English) Zbl 1466.49025

Summary: We prove the well-posedness and smoothing property of a fractional optimal control model with integral constraints governed by a hidden-memory variable-order Caputo time-fractional diffusion PDE, in which the adjoint equation leads to a different type of variable-order Riemann-Liouville time-fractional diffusion PDE. The L-1 discretization loses its monotonicity due to the impact of hidden memory, which was crucial in the error estimate of the L-1 discretization of constant-order fractional diffusion PDEs. We develop a novel splitting to prove an optimal-order error estimate of the discretization of the optimal control model without any artificial regularity assumption of the true solution. Numerical experiments are performed to substantiate the theoretical findings.

MSC:

49K40 Sensitivity, stability, well-posedness
26A33 Fractional derivatives and integrals
35K20 Initial-boundary value problems for second-order parabolic equations
49K20 Optimality conditions for problems involving partial differential equations
65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
65M60 Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs
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