The numerical approximation of an optimal control problem governed by a semilinear parabolic equation and constrained by a bound on the spatial $L^1$-norm of the control at every instant of time is studied. Spatial discretizations of the controls by piecewise constant and continuous piecewise linear functions are investigated. Under finite element approximations, the sparsity properties of the continuous solutions are preserved in a natural way using piecewise constant approximations of the control, but suitable numerical integration of the objective functional and of the constraint must be used to keep the sparsity pattern when using spatially continuous piecewise linear approximations. We also obtain error estimates and finally present some numerical examples.

中文翻译：

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具有非光滑逐点积分控制约束的最优控制问题数值逼近的误差估计

研究了由半线性抛物方程控制并受控制在每个时刻的空间$L^1$-范数上的界限约束的最优控制问题的数值逼近。研究了通过分段常数和连续分段线性函数对控件的空间离散化。在有限元近似下，连续解的稀疏特性使用控制的分段常数近似以自然方式保留，但在使用空间连续时，必须使用目标泛函和约束的适当数值积分来保持稀疏模式分段线性近似。我们还获得了误差估计，最后给出了一些数值例子。