次凸约束的多项式壳和H_∞控制
Polynomial hulls and H-infinity control for a hypoconvex constraint
摘要：
我们说C^n的一个子集是亚凸的，如果它的补是复超平面的并。设D是C中的封闭单位圆，T是单位圆.我们证明了赫尔顿和马歇尔的两个假设。(See"频域设计和分析选择“，印第安纳大学数学杂志，39，第1期（1990），157-184。设p：T × C^n --> R+是T上的子水平集具有紧次凸纤维的光滑函数.然后，在对p的某些限制条件下，若Y是p小于等于1的集合，则Y的多项式凸包是在Y中具有边界的解析向量值函数的图的并.进一步，设t是最小实数，使得p小于或等于t的集合包含圆盘上某解析向量值函数的图的边界。则只有一个解析向量值函数f使得p（z，f（z））对T中的所有z都小于或等于t.我们证明了f在T上是光滑的。我们还证明了，如果p关于一个参数平滑变化，那么唯一的f也是如此。
英文摘要：
We say that a subset of C^n is hypoconvex if its complement is the union of complex hyperplanes. Let D be the closed unit disk in C, T the unit circle. We prove two conjectures of Helton and Marshall. (See ``Frequency domain design and analytic selections,'' Indiana Univ. Math. J. 39, no. 1 (1990), 157-184.) Let p:T X C^n --> R+ be a smooth function whose sublevel sets have compact hypoconvex fibers over T. Then, with some restrictions on p, if Y is the set where p is less than or equal to 1, the polynomial convex hull of Y is the union of graphs of analytic vector-valued functions with boundary in Y. Furthermore, let t be the smallest real number such that the set where p is less than or equal to t contains the boundary of the graph of some analytic vector-valued function on the disk. Then there is only one analytic vector-valued function f such that p(z,f(z)) is less than or equal to t for all z in T. We show that f is smooth on T. We also prove that if p varies smoothly with respect to a parameter, so does the unique f just found.
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