Differentiability properties of the L1-tracking functional and application to the Robin inverse problem
Differentiability properties of the L1-tracking functional and application to the Robin inverse problem
复制标题
DOI：
10.1088/0266-5611/20/4/006
复制
发表时间：
2004-08
期刊：
影响因子：
2.1
通讯作者：
S. Chaabane;J. Ferchichi;K. Kunisch
中科院分区：
文献类型：
--
作者：
S. Chaabane;J. Ferchichi;K. Kunisch
我们研究了一个非标准形式的优化问题（OP）：代价泛函度量了直接Robin问题的解u与函数f L1（M）之间的L1距离。在证明了状态u关于的正性、单调性和控制性质后，我们证明了最优控制的存在性。问题（OP）并建立泛函的牛顿可微性。作为该优化问题的应用，考虑通过测量M上的数据f来确定Robin参数inv的逆问题。在这种情况下，f被假定为M上的迹。尽管事实上，我们的工作与L1-范数，我们证明了微分的成本功能，通过使用复杂的分析技术。证明是强相关的积极性和单调性的导数的状态。一个可识别的结果也证明了一组容许参数？AD由L？中的正函数组成。
We investigate an optimization problem (OP) in a non-standard form: the cost functional measures the L1 distance between the solution u of the direct Robin problem and a function f L1(M). After proving positivity, monotonicity and control properties of the state u with respect to , we prove the existence of an optimal control ? to the problem (OP) and establish Newton differentiability of the functional . As an application to this optimization problem the inverse problem of determining a Robin parameter inv by measuring the data f on M is considered. In that case f is assumed to be the trace on M of . In spite of the fact that we work with the L1-norm we prove differentiability of the cost functional by using complex analysis techniques. The proof is strongly related to positivity and monotonicity of the derivative of the state with respect to . An identifiability result is also proved for the set of admissible parameters ?ad consisting of positive functions in L?.