<p>This paper investigates adaptive finite element approximation of fractional semilinear optimal control problems involving the integral fractional Laplacian. The optimality system, derived from first-order necessary conditions, couples a semilinear fractional state equation, a linear fractional adjoint equation, and a control inclusion. The inherent nonlocality of fractional operators and the nonlinearity of the state equation pose significant challenges for efficient numerical approximation such as reduced convergence rates due to the lack of boundary regularity and the non-convexity. We consider two finite element discrete schemes: The state and adjoint state are both discretized by piecewise linear functions, while the control variable is approximated by variational discretization and piecewise constant functions, respectively. To effectively resolve the typically singular or localized behavior of the solutions near boundaries or specific points, we construct reliable residual-based a posteriori error estimators based on the estimate of Scott-Zhang operator in fractional Sobolev space and the property of first derivative of the reduced cost functional. These estimators quantify the discretization error across the state, adjoint, and control variables, driving local mesh refinement. Numerical experiments confirm the theoretical findings and demonstrate the superior efficiency of the adaptive approach compared to uniform mesh refinement.</p>

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Adaptive Finite Element Approximation of Semilinear Fractional Optimal Control Problem

  • Tongxin Wang,
  • Zhaojie Zhou

摘要

This paper investigates adaptive finite element approximation of fractional semilinear optimal control problems involving the integral fractional Laplacian. The optimality system, derived from first-order necessary conditions, couples a semilinear fractional state equation, a linear fractional adjoint equation, and a control inclusion. The inherent nonlocality of fractional operators and the nonlinearity of the state equation pose significant challenges for efficient numerical approximation such as reduced convergence rates due to the lack of boundary regularity and the non-convexity. We consider two finite element discrete schemes: The state and adjoint state are both discretized by piecewise linear functions, while the control variable is approximated by variational discretization and piecewise constant functions, respectively. To effectively resolve the typically singular or localized behavior of the solutions near boundaries or specific points, we construct reliable residual-based a posteriori error estimators based on the estimate of Scott-Zhang operator in fractional Sobolev space and the property of first derivative of the reduced cost functional. These estimators quantify the discretization error across the state, adjoint, and control variables, driving local mesh refinement. Numerical experiments confirm the theoretical findings and demonstrate the superior efficiency of the adaptive approach compared to uniform mesh refinement.