拟线性抛物方程的最优控制及其时间离散化
Optimal control of a quasilinear parabolic equation and its time discretization
AI总结：
本文研究拟线性抛物方程的最优控制，基于梯度流模型，通过隐式时间离散化，证明了控制到状态算子的存在性、Lipschitz连续性及全局极小值的存在性，并给出离散最优控制的收敛性。
摘要：
本文讨论了一类拟线性抛物状态方程的最优控制问题。该方程的形式源于例如控制各向异性Allen-Cahn方程作为晶体生长模型时所出现的问题类型。受此应用启发，我们将状态方程视为能量泛函的梯度流的结果。拟线性项是强单调的，并满足某种增长条件，而低阶项是非单调的。状态方程在时间上使用分片常数函数进行隐式离散。对于时间离散问题以及时间连续问题，均证明了控制到状态算子的存在性及其Lipschitz连续性。后者基于离散解的收敛性证明。最后，我们针对两者证明了全局极小值的存在性。此外，还可以证明时间离散最优控制的一个子序列收敛到时间连续问题的全局极小值。我们的结果在任意空间维度下均成立。
英文摘要：
In this paper we discuss the optimal control of a quasilinear parabolic state equation. Its form is leaned on the kind of problems arising for example when controlling the anisotropic Allen-Cahn equation as a model for crystal growth. Motivated by this application we consider the state equation as a result of a gradient flow of an energy functional. The quasilinear term is strongly monotone and obeys a certain growth condition and the lower order term is non-monotone. The state equation is discretized implicitly in time with piecewise constant functions. The existence of the control-to-state operator and its Lipschitz-continuity is shown for the time discretized as well as for the time continuous problem. Latter is based on the convergence proof of the discretized solutions. Finally we present for both the existence of global minimizers. Also convergence of a subsequence of time discrete optimal controls to a global minimizer of the time continuous problem can be shown. Our results hold in arbitrary space dimensions.
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