On Injectivity of an Integral Operator Connected to Riemann Hypothesis

Using the equivalent formulation of RH given by Beurling ([4],1955), Alcantara-Bode showed ([2], 1993) that Riemann Hypothesisholds if and only if the integral operator on the Hilbert space L2(0; 1)having the kernel defined by fractional part function of the expressionbetween brackets {y/x}, is injective.Since then, the injectivity of the integral operator used in equivalentformulation of RH has not been addressed nor has been dissociatedfrom RH and, a pure mathematics solution for RH is not ready yet.Here is a numerical analysis approach of the injectivity of the linearbounded operators on separable Hilbert spaces addressing the problemslike the one presented in [2]. Apart of proving the injectivity of theBeurling - Alcantara-Bode integral operator, we obtained the followingresult: every linear bounded operator (or its associated Hermitian),strict positive definite on a dense family of including approximationsubspaces in L2(0,1) built on simple functions, is injective if the rateof convergence to zero of its unbounded sequence of inverse conditionnumbers on approximation subspaces is o(n-s) for some s ≥ 0. Whens = 0, the sequence is inferior bounded by a not null constant, that isthe case in the Beurling - Alcantara-Bode integral operator.In the Theorem 4.1 we addressed with numerical analysis toolsthe injectivity of the integral operator in [2] claiming that - even if asolution in pure mathematics is desired, together with the Theorem 1,pg. 153 in [2], the RH holds.

H∞ Optimal Performance Design of an Unstable Plant under Bode Integral Constraint

This paper proposed the H∞ state feedback and H∞ output feedback design methods for unstable plants, which improved the original H∞ state feedback and H∞ output feedback. For the H∞ state feedback design of unstable plants, it presents the complete robustness constraint which is based on solving Riccati equation and Bode integral. For the H∞ output feedback design of unstable plants, the medium-frequency band should be considered in particular. Besides, this paper presents the method to select weight function or coefficients in the H∞ design, which employs Bode integral to optimize the H∞ design. It takes a magnetic levitation system as an example. The simulation results demonstrate that the optimal performance of perturbation suppression is obtained with the design of robustness constraint. The presented method is of benefit to the general H∞ design.

MOBIUS AND DELTA TRANSFORMS IN THE UNIFICATION OF CONTINUOUS-DISCRETE SPACES

It is well-known that in control theory the stability region of continuous- time system is laid in the left half plane of complex space, while that of discrete-time system is dwelled inside a unit circle. The former fact might be shown by exploiting the Laplace transform and the later by utilizing the corresponding zeta transform. In this paper we revealed the connectivity of both regions by employing M¨obius transform. We also used the same transform to derive continuous/discrete-time counterpart of several existing results, including Bode integral and Poisson-Jensen formula. We then demonstrated their unification property by using delta transform. Some numerical examples were provided to verify our results.

Bode Integral Constraints and the Zero Spillover Controller in Active Noise and Vibration Control

With the success of feedforward techniques, feedback control researchers have begun to explore the relationship between these two control paradigms. The goal of the present paper is to further investigate the relationship between feedforward and feedback control techniques by means of the classical Bode integral constraint on achievable performance. These results provide insight into the phenomenon of spillover which has been widely discussed in the vibration control literature. Specifically, it is shown that a particular feedforward controller called the zero spillover controller avoids spillover by producing perfect disturbance cancellation at every frequency. For readability, we derive the approximate zero spillover controller which is shown to be an optimal feedback controller for an LQG problem with suitable cross weights. Finally, the results are illustrated by means of analytical and numerical examples.