Strong stationarity for non-smooth control problems with fractional semi-linear elliptic equations in dimension $$N\le 3$$

Abstract In this paper, we investigate the optimal control of a semi-linear fractional PDEs involving the spectral diffusion operator, or the realization of the integral fractional Laplace operator with the zero Dirichlet exterior condition, both of order s with $$s\in (0,1)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>s</mml:mi> <mml:mo>∈</mml:mo> <mml:mo>(</mml:mo> <mml:mn>0</mml:mn> <mml:mo>,</mml:mo> <mml:mn>1</mml:mn> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> . The state equation contains a non-smooth nonlinearity, and the objective functional is convex in the control variable but contains non-smooth terms. As the mappings involved may not be Gâteaux differentiable, we use a regularization technique to regularize these nonlinear terms, aiming to obtain Gâteaux differentiable mappings. By employing this regularization technique, we are able to derive the first-order optimality condition for the regularized control problem by using the associated adjoint system. Furthermore, we conduct a limit analysis on the regularized term resulting in an optimality system for the non-smooth problem of C-stationary type. Subsequently, we establish a primal optimality condition, specifically B-stationarity. Under the assumption of “constraint qualification”, we derive the strong stationarity conditions for the non-smooth optimization problem with control constraints and establish the equivalence between B-stationarity and strong stationarity conditions.