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In this paper, we consider the optimal control of semilinear fractional PDEs with both spectral and integral fractional diffusion operators of order 2 s with s ∈ (0, 1). We first prove the boundedness of solutions to both semilinear fractional PDEs under minimal regularity assumptions on domain and data. We next introduce an optimal growth condition on the nonlinearity to show the Lipschitz continuity of the solution map for the semilinear elliptic equations with respect to the data. We further apply our ideas to show existence of solutions to optimal control problems with semilinear fractional equations as constraints. Under the standard assumptions on the nonlinearity (twice continuously differentiable) we derive the first and second order optimality conditions.
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This content will become publicly available on November 1, 2025
Optimal control of variably distributed‐order time‐fractional diffusion equation: Analysis and computation
Abstract Fractional diffusion equations exhibit competitive capabilities in modeling many challenging phenomena such as the anomalously diffusive transport and memory effects. We prove the well‐posedness and regularity of an optimal control of a variably distributed‐order fractional diffusion equation with pointwise constraints, where the distributed‐order operator accounts for, for example, the effect of uncertainties. We accordingly develop and analyze a fully‐discretized finite element approximation to the optimal control without any artificial regularity assumption of the true solution. Numerical experiments are also performed to substantiate the theoretical findings.
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- Award ID(s):
- 2245097
- PAR ID:
- 10629314
- Publisher / Repository:
- Elsevier
- Date Published:
- Journal Name:
- Numerical Methods for Partial Differential Equations
- Volume:
- 40
- Issue:
- 6
- ISSN:
- 0749-159X
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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