We adopt the integral definition of the fractional Laplace operator and study, on Lipschitz domains, an optimal control problem that involves a fractional elliptic partial differential equation (PDE) as state equation and a control variable that enters the state equation as a coefficient; pointwise constraints on the control variable are considered as well. We establish the existence of optimal solutions and analyze first and, necessary and sufficient, second order optimality conditions. Regularity estimates for optimal variables are also analyzed. We devise two strategies of finite element discretization: a semidiscrete scheme where the control variable is not discretized and a fully discrete scheme where the control variable is discretized with piecewise constant functions. For both solution techniques, we analyze convergence properties of discretizations and derive error estimates.
翻译:我们采用分解拉普特操作员的整体定义,并研究利普施奇茨域,这是一个最佳控制问题,它涉及作为州方程和将国家方程作为系数纳入国家方程的控制变量的分数等分部分异方程(PDE);也考虑对控制变量的点性限制;我们确定存在最佳解决方案,并分析第一和足够、必要和足够的第二顺序最佳条件;还分析最佳变量的规律性估计值。我们设计了两种有限元素分解战略:一种半分解方案,即控制变量不离散,另一种完全分离方案,控制变量与按片常态函数分离。对于这两种解决方案技术,我们分析离散的趋同特性并得出误差估计值。