2020/08/10 by Anne-Therese Rauls, Rauls, Anne-Therese, Stefan Ulbrich +1
Computer Science · Mathematics · #47J20 #49J40 #49J52 #49K40 #58C20 #58E35 #Analysis of PDEs (math.AP) #Contact Mechanics and Variational Inequalities #FOS: Mathematics #Numerical methods in inverse problems #Optimization and Control (math.OC) #Point processes and geometric inequalities
paper · pdf · doi:10.48550/arxiv.2008.04358
openalex publication_date 2020/08/10 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We consider optimal control problems for a wide class of bilateral obstacle\nproblems where the control appears in a possibly nonlinear source term. The\nnon-differentiability of the solution operator poses the main challenge for the\napplication of efficient optimization methods and the characterization of\nBouligand generalized derivatives of the solution operator is essential for\ntheir theoretical foundation and numerical realization. In this paper, we\nderive specific elements of the Bouligand generalized differential if the\ncontrol operator satisfies natural monotonicity properties. We construct\nmonotone sequences of controls where the solution operator is G ateaux\ndifferentiable and characterize the corresponding limit element of the\nBouligand generalized differential as being the solution operator of a\nDirichlet problem on a quasi-open domain. In contrast to a similar recent\nresult for the unilateral obstacle problem [RU19], we have to deal with an\nopposite monotonic behavior of the active and strictly active sets\ncorresponding to the upper and lower obstacle. Moreover, the residual is no\nlonger a nonnegative functional on H-1 and its representation as the\ndifference of two nonnegative Radon measures requires special care. This\nnecessitates new proof techniques that yield two elements of the Bouligand\ngeneralized differential. Also for the unilateral case we obtain an additional\nelement to that derived in [RU19].\n