arXiv subjects

Jun Zheng

Publications and source records attributed to Jun Zheng.

62 records · Page 4Linked to original sources

arXiv2018

Regularity in the two-phase free boundary problems under non-standard growth conditions

In this paper, we prove several regularity results for the heterogeneous, two-phase free boundary problems $\mathcal {J}_γ(u)=\int_Ω\big(f(x,\nabla u)+λ_{+} (u^{+})^γ+λ_{-}(u^{-})^γ+gu\big)\text{d}x\rightarrow \text{min}$ under non-standard growth conditions. Included in such problems are heterogeneous jets and cavities of Prandtl-Batchelor type with $γ=0$, chemical reaction problems with $0<γ<1$, and obstacle type problems with $γ=1$. Our results hold not only in the degenerate case of $p> 2$ for $p-$Laplace equations, but also in the singular case of $1<p<2$, which are extensions of \cite{LdT}.

math.AP↗

arXiv2017

Input-to-State Stability with Respect to Boundary Disturbances for a Class of Semi-linear Parabolic Equations

This paper studies the input-to-state stability (ISS) properties based on the method of Lyapunov functionals for a class of semi-linear parabolic partial differential equations (PDEs) with respect to boundary disturbances. In order to avoid the appearance of time derivatives of the disturbances in ISS estimates, some technical inequalities are first developed, which allow directly dealing with the boundary conditions and establishing the ISS based on the method of Lyapunov functionals. The well-posedness analysis of the considered problem is carried out and the conditions for ISS are derived. Two examples are used to illustrate the application of the developed result.

math.OC↗

arXiv2017

A De Giorgi Iteration-based Approach for the Establishment of ISS Properties of a Class of Semi-linear Parabolic PDEs with Boundary and In-domain Disturbances

This paper addresses input-to-state stability (ISS) properties with respect to boundary and in-domain disturbances for a class of semi-linear partial differential equations (PDEs) subject to Dirichlet boundary conditions. The developed approach is a combination of the method of De Giorgi iteration and the technique of Lyapunov functionals by adequately splitting the original problem into two subsystems. The ISS in $L^\infty$-norm for a 1-$D$ transport equation and the ISS in $L^2$-norm for Burgers' equation have been established using this method. As an application of the main result for the 1-D transport equation, a study on ISS properties in $L^\infty$-norm of a 1-D transport equation with anti-stable term under boundary feedback control is carried out. This is the first time that the method of De Giorgi iteration is introduced in the ISS theory for infinite dimensional systems, and the developed method can be generalized for tackling some problems on multidimensional spatial domains and be applied to a wider class of nonlinear parabolic PDEs.

math.OC↗

arXiv2017

Distributed Weighted Sum-Rate Maximization in Multicell MU-MIMO OFDMA Downlink

This paper considers distributed linear beamforming in downlink multicell multiuser orthogonal frequency-division multiple access networks. A fast convergent solution maximizing the weighted sum- rate with per base station (BS) transmiting power constraint is formulated. We approximate the non- convex weighted sum-rate maximization (WSRM) problem with a semidefinite relaxed solvable convex form by means of a series of approximation based on interference alignment (IA) analysis. The WSRM optimization is a two-stage optimization process. In the first stage, the IA conditions are satisfied. In the second stage, the convex approximation of the non-convex WSRM is obtained based on the consequences of IA, and high signal-to-interference-plus-noise ratio assumption. Compared to the conventional iterative distributed algorithms where the BSs exchange additional information at each iteration, the BSs of our proposed solution optimize their beamformers locally without reporting additional information during the iterative procedure.

cs.IT↗

arXiv2015

The obstacle problem for nonlinear degenerate equations with $L^{1}$-data

The aim of this paper is to study the obstacle problem with an elliptic operator having degenerate coercivity. We prove the existence of an entropy solution to the obstacle problem under the assumption of $L^{1}-$summability on the data. Meanwhile, we prove that every entropy solution belongs to some Sobolev space $W^{1,q}(Ω)$.

math.AP↗

arXiv2015

Flatness-based Deformation Control of an Euler-Bernoulli Beam with In-domain Actuation

This paper addresses the problem of deformation control of an Euler-Bernoulli beam with in-domain actuation. The proposed control scheme consists in first relating the system model described by an inhomogeneous partial differential equation to a target system under a standard boundary control form. Then, a combination of closed-loop feedback control and flatness-based motion planning is used for stabilizing the closed-loop system around reference trajectories. The validity of the proposed method is assessed through well-posedness and stability analysis of the considered systems. The performance of the developed control scheme is demonstrated through numerical simulations of a representative micro-beam.

math.OC↗

arXiv2015

In-domain control of a heat equation: an approach combining zero-dynamics inverse and differential flatness

This paper addresses the set-point control problem of a heat equation with in-domain actuation. The proposed scheme is based on the framework of zero dynamics inverse combined with flat system control. Moreover, the set-point control is cast into a motion planing problem of a multiple-input, multiple-out system, which is solved by a Green's function-based reference trajectory decomposition. The validity of the proposed method is assessed through convergence and solvability analysis of the control algorithm. The performance of the developed control scheme and the viability of the proposed approach are confirmed by numerical simulation of a representative system.

math.OC↗

arXiv2015

Controlled Remote State Preparation via General Pure Three-Qubit State

The protocols for controlled remote state preparation of a single qubit and a general two-qubit state are presented in this paper. The general pure three-qubit states are chosen as shared quantum channel, which are not LOCC equivalent to the mostly used GHZ-state. It is the first time to introduce general pure three-qubit states to complete remote state preparation. The probability of successful preparation is presented. Moreover, in some special cases, the successful probability could reach unit.

quant-ph↗