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首页> 外文期刊>IEEE Transactions on Automatic Control >Operator theoretic approach to the optimal two-disk problem
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Operator theoretic approach to the optimal two-disk problem

机译：最优两盘问题的算子理论方法

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摘要

The nonstandard two-disk problem plays a fundamental role in robust feedback optimization. Here, it is shown via Banach space duality theory that its solutions satisfy an extremal identity, and may be viewed as a dual extremal kernel of a particular L1-optimization problem. A novel operator theoretic framework to characterize explicitly its solutions is developed, in particular, the two-disk optimization is shown to be equal to the induced norm of a specific operator defined on a projective tensor product space involving a non-Hilbert version of a vector valued H2 space. Moreover, this operator is shown to be a combination of multiplication and Toeplitz operators. Under certain conditions, existence of maximal vectors is established leading to an explicit formula for the optimal controller. An "infinite matrix" representation with respect to a canonical basis is derived, together with an algorithm to compute it. The norm of the relevant operator is approximated by special finite dimensional optimizations whose solutions lead to solving semi-definite programming problems involving the computation of a matrix projective tensor norm.
机译：非标准的两盘问题在可靠的反馈优化中起着基本作用。在这里，通过Banach空间对偶性理论表明，其解满足极值恒等式，并且可以视为特定L1优化问题的对偶极值核。开发了一种新颖的算子理论框架来明确表征其解决方案，特别是，两盘优化被证明等于在涉及向量的非希尔伯特版本的投影张量积空间上定义的特定算子的诱导范数重视H2空间。此外，该运算符显示为乘法和Toeplitz运算符的组合。在某些条件下，建立了最大向量，从而为最优控制器建立了明确的公式。得出关于规范基础的“无限矩阵”表示形式，以及用于对其进行计算的算法。相关算子的范数通过特殊的有限维优化来近似，其解导致解决半定规划问题，涉及矩阵投影张量范数的计算。

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