Mixed sensitivity minimization problems with rationall 1-optimal solutions

Up to this time, the only known method to solve the discrete-time mixed sensitivity minimization problem inl ¹ has been to use a certain infinite-dimensional linear programming approach, presented by Dahleh and Pearson in 1988 and later modified by Mendlovitz. That approach does not give in general true optimal solutions; only suboptimal ones are obtained. Here, for the first time, the truel ¹-optimal solutions are found for some mixed sensitivity minimization problems. In particular, Dahleh and Pearson construct an 11h order suboptimal compensator for a certain second-order plan with first-order weight functions; it is shown that the unique optimal compensator for that problem is rational and of order two. The author discovered this fact when trying out a new scheme of solving the infinite-dimensional linear programming system. This scheme is of independent interest, because when it is combined with the Dahleh-Pearson-Mendlovitz scheme, it gives both an upper bound and a lower bound on the optimal performance; hence, it provides the missing error bound that enables one to truncate the solution. Of course, truncation is appropriate only if the order of the optimal compensator is too high. This may indeed be the case, as is shown with an example where the order of the optimal compensator can be arbitrarily high.     

平面上有限级Dirichlet级数和随机Dirichlet级数的增长性

利用型函数及Newton多边形讨论了平面上有限级Dirichlet级数和随机Dirichlet级数的增长性和系数间的关系。通过引理得出:当r=eσ(σ→+∞)时,Dirichlet级数的增长性和系数间的重要关系,以及对于随机变量序列{Xn}满足条件:存在α>0,使得supn 0E(|Xn|α)<∞;存在β>0,使得supn 0E(|Xn|-β)<∞的随机Dirichlet级数f(s,ω)=∞n=0bnXn(ω)eλns和Dirichlet级数f(s)=∞n=0bneλns有几乎相同的关于型函数的增长性。     

Tractability of quasilinear problems II: Second-order elliptic problems

In a previous paper, we developed a general framework for establishing tractability and strong tractability for quasilinear multivariate problems in the worst case setting. One important example of such a problem is the solution of the Helmholtz equation in the -dimensional unit cube, in which depends linearly on , but nonlinearly on . Here, both and  are -variate functions from a reproducing kernel Hilbert space with finite-order weights of order . This means that, although  can be arbitrarily large, and  can be decomposed as sums of functions of at most  variables, with independent of .

In this paper, we apply our previous general results to the Helmholtz equation, subject to either Dirichlet or Neumann homogeneous boundary conditions. We study both the absolute and normalized error criteria. For all four possible combinations of boundary conditions and error criteria, we show that the problem is tractable. That is, the number of evaluations of and  needed to obtain an -approximation is polynomial in  and , with the degree of the polynomial depending linearly on . In addition, we want to know when the problem is strongly tractable, meaning that the dependence is polynomial only in  , independently of . We show that if the sum of the weights defining the weighted reproducing kernel Hilbert space is uniformly bounded in  and the integral of the univariate kernel is positive, then the Helmholtz equation is strongly tractable for three of the four possible combinations of boundary conditions and error criteria, the only exception being the Dirichlet boundary condition under the normalized error criterion.

     

MIMOl 1-optimization with a scalar control

In Ref. 1, the author showed that some scalar mixed sensitivity minimization problems have rationall ¹-optimal solutions and computed these solutions. Here, a similar approach is applied to a multi-input multi-output (MIMO) plant. The general flavor of the solution is the same as in Ref. 1, but there is one distinct new feature. We prove that some of the outputs may be ignored, in the sense that they have no influence on the optimal solution and that all the operators corresponding to the remaining outputs have the same optimal norm. This fact suggests alternative methods for solving the problem and it is employed in the construction of an exact rationall ¹-optimal solution of a particular problem.This work was partly done while the author was a visiting professor at the Department of Mathematics, Virginia Polytechnic Institute and State University, Blacksburg, Virginia. Supported in part by a joint grant from NSF and the Academy of Finland.