ROBUSTNESS OF OPTIMALLY ROBUST CONTROLLERS——Dedicated to the Sixtieth Birthday of Prof. You Zhao-yong

The largest robust stability radius r(P0) of a system P0 is defined as the radius of the largest ball Bmax in the gap metric centered at P0 which can be stabilized by one single controller. Any controller which stabilizes Bmax is called an optimally robust controller of P0. Any controller, regarded as a system, should have its own largest robust stability radius also. In this paper it is shown that the largest robust stability radius of any optimally robust controller of P0 is larger than or equal to r(Po). Moreover, the variation of the closed-loop transfer matrix caused by the perturbation of the system is estimated.

Amplitude analysis of the decays D^(0)→π^(+)π^(−)π^(+)π^(−)and D^(0)→π^(+)π^(−)π^(0)π^(0)

Using e^(+)e^(−)annihilation data corresponding to an integrated luminosity of 2.93 fb^(−1)taken at the center-of-mass energy√s=3.773 GeV with the BESIII detector,a joint amplitude analysis is performed on the decays D^(0)→π^(+)π^(−)π^(+)π^(−)and D^(0)→π^(+)π^(−)π^(0)π^(0)(non-η).The fit fractions of individual components are obtained,and large interferences among the dominant components of the decays D^(0)→a_(1)(1260)π,D^(0)→π(1300)π,D^(0)→ρ(770)ρ(770),and D^(0)→2(ππ)_(S)are observed in both channels.With the obtained amplitude model,the CP-even fractions of D^(0)→π^(+)π^(−)π^(+)π^(−)and D^(0)→π^(+)π^(−)π^(0)π^(0)(non-η)are determined to be(75.2±1.1_(stat).±1.5_(syst.))%and(68.9±1.5_(stat).±2.4_(syst.))%,respectively.The branching fractions of D^(0)→π^(+)π^(−)π^(+)π^(−)and D^(0)→π^(+)π^(−)π^(0)π^(0)(non-η)are measured to be(0.688±0.010_(stat.)±0.010_(syst.))%and(0.951±0.025_(stat.)±0.021_(syst.))%,respectively.The amplitude analysis provides an important model for the binning strategy in measuring the strong phase parameters of D^(0)→4πwhen used to determine the CKM angleγ(ϕ_(3))via the B^(−)→DK^(−)decay.

Observation of e^(+)e^(-)→pppñπ-+c.c.

Using data taken at 29 center-of-mass energies between 4.16 and 4.70 GeV with the BESⅢdetector at the Beijing Electron Positron Collider corresponding to a total integrated luminosity of approximately 18.8 fb^(-1),the process e^(+)e^(-)→pppñπ+c.c.is observed for the first time with a statistical significance of 11.5σ.The average Born cross sections in the energy ranges of(4.160,4.380)GeV,(4.400,4.600)GeV and(4.610,4.700)GeV are measured to be(21.5±5.7±1.2)fb,(46.3±10.6±2.5)fb and(59.0±9.4±3.2)fb,respectively,where the first uncertainties are statistical and the second are systematic.The line shapes of the pñ and ppπ^(-)invariant mass spectra are consistent with phase space distributions,indicating that no hexaquark or di-baryon state is observed.

Search for hidden-charm tetraquark with strangeness in e^(+)e^(−)→K^(+)D_(s)^(∗−) D^(∗0 )+ c.c.

We report a search for a heavier partner of the recently observed Z_(cs)(3985)^(-) state,denoted as Z_(cs)^('-),in the process e^(+)e^(−)→K^(+)D_(s)^(∗−) D^(∗0 )+ c.c.,based on e^(*)e^(-)collision data collected at the center-of-mass energies of √s=4.661,4.682 and 4.699 GeV with the BESIII detector.The Z_(cs)^('-) is of interest as it is expected to be a candidate for a hidden-charm and open-strange tetraquark.A partial-reconstruction technique is used to isolate K^(+)recoil-mass spectra,which are probed for a potential contribution from Z_(cs)^('-)→D_(s)^(∗−) D^(∗0 )+ c.c.We find an excess of Z_(cs)^('-)→D_(s)^(*-)-D^(*0)(c.c.)candidates with a significance of 2.1o,after considering systematic uncertainties,at a mass of(4123.5±0.7_(sat)±4.7_(syst.))MeV/c^(2).As the data set is limited in size,the upper limits are evaluated at the 90%confidence level on the product of the Born cross sections(σ^(Borm))and the branching fraction(B)of Z_(cs)^('-)→D_(s)^(*-)-D^(*0),under different assumptions of the Z_(cs)^('-) mass from 4.120 to 4.140 MeV and of the width from 10 to 50 MeV at the three center-of-mass energies.The upper limits of σ^(Born).B are found to be at the level of O(1)pb at each energy.Larger data samples are needed to confirm the Z_(cs)^('-) state and clarify its nature in the coming years.

Search for the weak decayψ(3686)→■+c.c.

Using(448.1±2.9)×10^(6)ψ(3686)for the weak baryonic decayψ(3686)→Λc+∑-+c.c..The analysis procedure is optimized using a blinded method.No significant signal is observed,and the upper limit on the branching fraction(B)ofψ(3686)→Λc+∑-+c.c.is set as 1.4×10^(-5)at the 90%confidence level.

Observations of the Cabibbo-Suppressed decays Λ_(c)^(+)→nπ+π^(0),nπ^(+)π^(-)π^(+) and the Cabibbo-Favored decay Λ_(c)^(+)→nK^(-)π^(+)π^(+)

Using electron-positron annihilation data samples corresponding to an integrated luminosity of 4.5 fb-1,collected by the BESⅢdetector in the energy region between 4599.53 MeV and 4698.82 MeV,we report the first observations of the Cabibbo-suppressed decaysΛ_(c)^(+)→nπ^(+)π^(0),Λ_(c)^(+)→nπ^(+)π^(-)π^(+),and the Cabibbo-favored decayΛ_(c)^(+)→nK^(-)π^(+)π^(+)with statistical significances of 7.9σ,7.8σ,and>10σ,respectively.The branching fractions of these decays are measured to be B(Λ_(c)^(+)→nπ^(+)π^(0))=(0.64±0.09±0.02)%,B(Λ_(c)^(+)→nπ^(+)π^(-)π^(+))=(0.45±0.07±0.03)%,and B(Λ_(c)^(+)→nK^(-)π^(+)π^(+))=(1.90±0.08±0.09)%,where the first uncertainties are statistical and the second are systematic.We find that the branching fraction of the decayΛ_(c)^(+)→nπ^(+)π^(0)is about one order of magnitude higher than that ofΛ_(c)^(+)→nπ^(+).

Determination of the number ofψ(3686)events taken at BESⅢ

The number ofψ(3686)events collected by the BESⅢdetector during the 2021 run period is determined to be(2259.3±11.1)×10~6 by counting inclusiveψ(3686)hadronic events.The uncertainty is systematic and the statistical uncertainty is negligible.Meanwhile,the numbers ofψ(3686)events collected during the 2009 and 2012run periods are updated to be(107.7±0.6)×10~6 and(345.4±2.6)×10~6,respectively.Both numbers are consistent with the previous measurements within one standard deviation.The total number ofψ(3686)events in the three data samples is(2712.4±14.3)×10~6.