We study optimal design problems where the design corresponds to a coefficient in the principal part of the state equation. The state equation, in addition, is parameter dependent, and we allow it to change type in the limit of this (modeling) parameter. We develop a framework that guarantees asymptotic compatibility, that is unconditional convergence with respect to modeling and discretization parameters to the solution of the corresponding limiting problems. This framework is then applied to two distinct classes of problems where the modeling parameter represents the degree of nonlocality. Specifically, we show unconditional convergence of optimal design problems when the state equation is either a scalar-valued fractional equation, or a strongly coupled system of nonlocal equations derived from the bond-based model of peridynamics.

翻译：我们研究一类最优设计问题，其中设计变量对应于状态方程主部中的系数。此外，状态方程依赖于参数，且我们允许该方程在此（建模）参数的极限下发生类型转变。我们建立了一个保证渐近相容性的理论框架，即关于建模参数与离散化参数无条件收敛到相应极限问题的解。随后将该框架应用于两类不同的问题，其中建模参数表示非局部性的程度。具体而言，我们证明了当状态方程为标量值分数阶方程时，或为由基于键的近场动力学模型导出的强耦合非局部方程组时，最优设计问题均具有无条件收敛性。