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：本文考虑两类偏微分方程半线性最优控制系统的全离散有限元逼近：分布控制与Robin边界控制。采用协调有限元方法对生态捕食者-食饵最优控制模型的空间部分进行逼近，时间离散则采用间断伽辽金方法。我们研究了解与目标函数距离的敏感性，涵盖光滑与粗糙初始数据两种情形。在适当正则性条件下，采用空间与时间上的低阶和高阶多项式。所考虑的近似格式分为含控制约束与无控制约束两类，通过非线性梯度方法有效驱动系统达到期望状态。