Let Z=（ Zt）t≥0 be a Bessel process of dimension δ（ δ〉0） starting at zero and let K（t） be a differentiable function on [0,∞） with K（t）〉0 （A↓t≥0）. Then we establish the relationship between L^p-norm o...Let Z=（ Zt）t≥0 be a Bessel process of dimension δ（ δ〉0） starting at zero and let K（t） be a differentiable function on [0,∞） with K（t）〉0 （A↓t≥0）. Then we establish the relationship between L^p-norm of log^1/2（1 ＋δJτ） and L^p-norm of sup Zt[t＋k（t）]^-1/2 （0≤t≤τ） for all stopping times τ and all 0〈p〈＋∞.As an interesting example, we show that ｜｜log^1/2（1＋δLm＋1（τ）｜｜p and ｜｜supZtП[1＋Lj（t]^1/2｜｜p （0≤j≤m,j∈Z;0≤t≤τ） are equivalent with 0〈p〈＋∞ for all stopping times rand all integer numbers m, where the function Lm（t） （t≥0） is inductively defined by Lm＋1（t）=log[ 1 ＋Lm（t）] with L0（t）= 1.展开更多
This paper considers dynamical systems under feedback with control actions limited toswitching.The authors wish to understand the closed-loop systems as approximating multi-scale problemsin which the implementation of...This paper considers dynamical systems under feedback with control actions limited toswitching.The authors wish to understand the closed-loop systems as approximating multi-scale problemsin which the implementation of switching merely acts on a fast scale.Such hybrid dynamicalsystems are extensively studied in the literature,but not much so far for feedback with partial stateobservation.This becomes in particular relevant when the dynamical systems are governed by partialdifferential equations.The authors introduce an augmented BV setting which permits recognition ofcertain fast scale effects and give a corresponding well-posedness result for observations with such minimalregularity.As an application for this setting,the authors show existence of solutions for systemsof semilinear hyperbolic equations under such feedback with pointwise observations.展开更多
基金Project supported by the National Natural Science Foundation of China (No. 10571025) and the Key Project of Chinese Ministry of Education (No. 106076)
文摘Let Z=（ Zt）t≥0 be a Bessel process of dimension δ（ δ〉0） starting at zero and let K（t） be a differentiable function on [0,∞） with K（t）〉0 （A↓t≥0）. Then we establish the relationship between L^p-norm of log^1/2（1 ＋δJτ） and L^p-norm of sup Zt[t＋k（t）]^-1/2 （0≤t≤τ） for all stopping times τ and all 0〈p〈＋∞.As an interesting example, we show that ｜｜log^1/2（1＋δLm＋1（τ）｜｜p and ｜｜supZtП[1＋Lj（t]^1/2｜｜p （0≤j≤m,j∈Z;0≤t≤τ） are equivalent with 0〈p〈＋∞ for all stopping times rand all integer numbers m, where the function Lm（t） （t≥0） is inductively defined by Lm＋1（t）=log[ 1 ＋Lm（t）] with L0（t）= 1.
基金support of the Elite Network of Bavaria under the grant #K-NW-2004-143
文摘This paper considers dynamical systems under feedback with control actions limited toswitching.The authors wish to understand the closed-loop systems as approximating multi-scale problemsin which the implementation of switching merely acts on a fast scale.Such hybrid dynamicalsystems are extensively studied in the literature,but not much so far for feedback with partial stateobservation.This becomes in particular relevant when the dynamical systems are governed by partialdifferential equations.The authors introduce an augmented BV setting which permits recognition ofcertain fast scale effects and give a corresponding well-posedness result for observations with such minimalregularity.As an application for this setting,the authors show existence of solutions for systemsof semilinear hyperbolic equations under such feedback with pointwise observations.
