  In this paper we present and analyze a weighted residual a posteriori error
estimate for an optimal control problem. The problem involves a
nondifferentiable cost functional, a state equation with an integral fractional
Laplacian, and control constraints. We employ subdifferentiation in the context
of nondifferentiable convex analysis to obtain first-order optimality
conditions. Piecewise linear polynomials are utilized to approximate the
solutions of the state and adjoint equations. The control variable is
discretized using the variational discretization method. Upper and lower bounds
for the a posteriori error estimate of the finite element approximation of the
optimal control problem are derived. In the region where 3/2 < alpha < 2, the
residuals do not satisfy the L2(Omega) regularity. To address this issue, an
additional weight is included in the weighted residual estimator, which is
based on a power of the distance from the mesh skeleton. Furthermore, we
propose an h-adaptive algorithm driven by the posterior view error estimator,
utilizing the Dorfler labeling criterion. The convergence analysis results show
that the approximation sequence generated by the adaptive algorithm converges
at the optimal algebraic rate. Finally, numerical experiments are conducted to
validate the theoretical results.
