A DUALITY-BASED APPROACH TO ELLIPTIC CONTROL PROBLEMS IN NON-REFLEXIVE BANACH SPACES
A DUALITY-BASED APPROACH TO ELLIPTIC CONTROL PROBLEMS IN NON-REFLEXIVE BANACH SPACES
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DOI：
10.1051/cocv/2010003
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发表时间：
2011-01-01
影响因子：
1.4
通讯作者：
Kunisch, Karl
中科院分区：
文献类型：
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作者：
Clason, Christian;Kunisch, Karl
凸对偶是解决非光滑最优控制问题的有力框架。然而，对于L-1（Omega）或BV（Omega）等非自反巴拿赫空间中的问题，对偶问题是在具有难以测量的理论结构的空间中表述的。另一方面，预预问题可以在Hilbert空间中表述，并且包含具有盒形约束的光滑泛函的最小化，并且存在有效的数值方法。本文研究了以有界变分函数和测度为控制的椭圆控制问题。讨论了相应前偶问题的存在唯一性，并用半光滑牛顿法求解了最优性系统。数值例子说明了这些巴拿赫空间中最优控制的结构差异，与在相应的希尔伯特空间设置中获得的最优控制相比。
Convex duality is a powerful framework for solving non-smooth optimal control problems. However, for problems set in non-reflexive Banach spaces such as L-1(Omega) or BV(Omega), the dual problem is formulated in a space which has difficult measure theoretic structure. The predual problem, on the other hand, can be formulated in a Hilbert space and entails the minimization of a smooth functional with box constraints, for which efficient numerical methods exist. In this work, elliptic control problems with measures and functions of bounded variation as controls are considered. Existence and uniqueness of the corresponding predual problems are discussed, as is the solution of the optimality systems by a semismooth Newton method. Numerical examples illustrate the structural differences in the optimal controls in these Banach spaces, compared to those obtained in corresponding Hilbert space settings.