Geometric characterization on the solvability of regulator equations

The solvability of the regulator equation for a general nonlinear system is discussed in this paper by using geometric method. The ‘feedback’ part of the regulator equation, that is, the feasible controllers for the regulator equation, is studied thoroughly. The concepts of minimal output zeroing control invariant submanifold and left invertibility are introduced to find all the possible controllers for the regulator equation under the condition of left invertibility. Useful results, such as a necessary condition for the output regulation problem and some properties of friend sets of controlled invariant manifolds, are also obtained

Radiation-driven Implosion in the Cepheus B Molecular Cloud

We analyze large scale mapping observations of the molecular lines in the 12CO (J=2-1), 12CO (J=3-2), 13CO (J=2-1), and 13CO (J=3-2) transition emissions toward the Cepheus B molecular cloud with the KOSMA 3m-telescope. The integrated intensity map of the 12CO (J=2-1) transition has shown a structure with a compact core and a compact ridge extended in the north-west of the core. The cloud is surrounded by an optically bright rim, where the radiation-driven implosion (RDI) may greatly change the gas properties. The intensities of the CO (J=3-2) transition are higher than those of the CO (J=2-1) transition along the rim area.We find characteristic RDI structure in positionvelocity diagrams. Non-LTE Large velocity gradient (LVG) model analysis shows that the density and temperature at the edge are higher than that in the center. Our results provide evidences that RDI is taking place in Cepheus B molecular cloud.Comment: 8 pages, 5 figure

Nonconforming Virtual Element Method for 2m2m-th Order Partial Differential Equations in Rn\mathbb R^n

A unified construction of the $H^m$-nonconforming virtual elements of any order $k$ is developed on any shape of polytope in $\mathbb R^n$ with constraints $m\leq n$ and $k\geq m$. As a vital tool in the construction, a generalized Green's identity for $H^m$ inner product is derived. The $H^m$-nonconforming virtual element methods are then used to approximate solutions of the $m$-harmonic equation. After establishing a bound on the jump related to the weak continuity, the optimal error estimate of the canonical interpolation, and the norm equivalence of the stabilization term, the optimal error estimates are derived for the $H^m$-nonconforming virtual element methods.Comment: 33page