We consider fully discrete finite element approximations for a semilinear optimal control system of partial differential equations in two cases: for distributed and Robin boundary control. The ecological predator-prey optimal control model is approximated by conforming finite element methods mimicking the spatial part, while a discontinuous Galerkin method is used for the time discretization. We investigate the sensitivity of the solution distance from the target function, in cases with smooth and rough initial data. We employ low, and higher-order polynomials in time and space whenever proper regularity is present. The approximation schemes considered are with and without control constraints, driving efficiently the system to desired states realized using non-linear gradient methods.

翻译：我们考虑两类半线性偏微分方程最优控制系统的全离散有限元逼近：分布式控制与Robin边界控制。生态捕食者-猎物最优控制模型采用空间部分一致的协调有限元方法进行逼近，而时间离散则使用间断伽辽金方法。我们研究了在光滑与粗糙初值情形下，解与目标函数之间的灵敏度。在具备适当正则性时，我们采用时间与空间上的低阶及高阶多项式。所考虑的逼近方案包含或不包含控制约束，能够通过非线性梯度方法高效地将系统驱动至期望状态。