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Wed, 20 Aug 2014 11:13:55 GMT2014-08-20T11:13:55ZStability of the stationary solution to Vlasov-Poisson-Boltzmann system with hard potential
http://dspace.cityu.edu.hk:80/handle/2031/7308
Title: Stability of the stationary solution to Vlasov-Poisson-Boltzmann system with hard potential
Authors: Wang, Lusheng (王路生)
Abstract: The Vlasov-Poisson-Boltzmann (VPB) system is a physical model describing
the time evolution of number density distribution of charged dilute gases.
The self-consistent potential generating the electric field is related to the
unknown distribution function by the Poisson equation. There is extensive
research on the VPB system and some new phenomena in the long time
behaviour have been discovered.
However, most of the previous research on the Cauchy problem is about
the hard sphere case, and the difficulty of extending the results to the hard
potential case lies in controlling the large-velocity growth in the nonlinear
term. Recently, there are some studies about this hard potential case with
constant background density and the analysis is based on introducing a new
time-velocity weight function in the energy functional and its dissipation
to control the large growth. In this thesis, we consider the case for hard
potential with stationary background density.
The commonly used perturbation scheme leads to an equation with a term
involving no dissipation. Hence, it's very difficult to control. We deal with
this term by introducing a new perturbation of the stationary solution. Then
by multiplying the collision invariants, we derive the fluid-type equations for
the macroscopic parts of the solution. Based on the resulting equations and
an a priori assumption about the decay rate of the self-consistent electric
potential, we obtain the energy estimates for the solution and its dissipation
rate given by the microscopic parts. We also need the macroscopic dissipation
rate to absorb the right hand terms appearing in the previous estimates. To
do this, we show the existence of an interactive energy functional and its
dissipation rate contains macroscopic parts of the solution.
In order to absorb one term arising from the a priori assumption in the
above estimates, we need the weighted energy estimates. Using induction
and the linear combination of the previous non-weighted energy estimates,
we obtain the desired estimates and its dissipation given by the full weighted
dissipation rate.
Subsequently, we show the local existence of the unique solution under
the natural mass conservation condition and two a priori assumptions. The
analysis is based on the previously obtained weighted energy estimates, the
high order energy estimates and time decay estimate for a linearized system.
Finally, we obtain the uniform in time estimates of the solution by the continuity
argument and show that the a priori assumptions we made can be
closed.
Notes: CityU Call Number: QC175.2 .W36 2012; 93 leaves 30 cm.; Thesis (Ph.D.)--City University of Hong Kong, 2012.; Includes bibliographical references (leaves 89-93)Sun, 01 Jan 2012 00:00:00 GMThttp://dspace.cityu.edu.hk:80/handle/2031/73082012-01-01T00:00:00ZNetwork-based stability analysis and synthesis for stochastic system with Markovian switching
http://dspace.cityu.edu.hk:80/handle/2031/7307
Title: Network-based stability analysis and synthesis for stochastic system with Markovian switching
Authors: Liu, Ming (劉明)
Abstract: This thesis is concerned with the network-based stability analysis and synthesis
of stochastic systems with Markovian characteristics. Two difficulties raised in
Networked Control Systems (NCSs) are considered: Random Network Communication
Delay and Quantization (state quantization and input quantization are
discussed respectively). Four types of research problems are investigated. They
are:
1. Stabilization of Markovian jump linear system over networks with random
communication delay.
2. Logarithmic Quantizer design for Markovian jump linear system via Input
Partition method and via State Partition Method.
3. Quantization error estimation and observer-based controller design.
4. Robust quantized filtering for Ito stochastic system with Markvian switching.
It is well known that, the network communication transmission delay is a
common phenomenon in NCSs. In Problem 1, the main work is concerned with
the stabilization problem for a networked control system with Markovian characterization
in the existence of random communication delay. We consider the case
that the random communication delays exist both in the system state and in the
mode signal which are modelled as a Markovian chain. The resulting closed-loop
system is modelled as a Markovian jump linear system with two jumping parameters,
and a necessary and sufficient condition on the existence of stabilizing
controllers is established.
The state/input quantization is another important phenomenon in NCSs. In
Problem 2, the input quantization is addressed. We shall first deal with the
problem by designing a quantized control strategy for a class of discrete time
Markovian jump linear systems using input partition method. We generalize the
classic static quantizer design method to Markovian jump linear system. The concept
of mode-dependent logarithmic quantizer is presented, which is employed to
ensure the stochastic stability of the quantized closed-loop system. Secondly,
in Problem 2, we aim to design another quantized control strategy for discrete
time Markovian jump linear systems via state partition method. We introduce
the concept of stochastically quadratically stabilizing a mode-dependent quantizer
associated with a series of new definitions. The proposed mode-dependent
quantizer can stabilize the quantized closed-loop system and overcome the effect
of Markovian switching on system stability.
In Problem 3, we shall discuss the problem of state quantization. Most of the
quantized control schemes in existing literature are focused on the controller/filter
design using the quantized state/output signal. Since error always exists between
the quantized signal and the original one, the designed controller/filter in the
framework of traditional quantized control scheme may not achieve an ideal performance.
Hence, we consider a quantized control strategy by decoupling the
original state/output from the quantized signal by using the observer technique.
A quantization-tolerant controller is then designed to stabilize the closed-loop
quantized system. We shall employ this control strategy to deal with the stabilization
of continuous linear systems with limited communication capacity.
In Problem 4, we investigate a filtering design for uncertain stochastic systems
with saturating quantized measurements. There are time-varying parameter uncertainties,
and state and external-disturbance-dependent noise in the plant. In
the presence of output quantization, we consider the design of robust quantized
filters for Ito stochastic systems, and sufficient conditions are obtained such that
the filtering error systems are robustly exponentially stable.
Notes: CityU Call Number: QA274.7 .L58 2009; viii, 166 leaves : ill. 30 cm.; Thesis (Ph.D.)--City University of Hong Kong, 2009.; Includes bibliographical references (leaves [152]-166)Thu, 01 Jan 2009 00:00:00 GMThttp://dspace.cityu.edu.hk:80/handle/2031/73072009-01-01T00:00:00ZSome nonlinear problems on manifolds arising from conformal geometry
http://dspace.cityu.edu.hk:80/handle/2031/7213
Title: Some nonlinear problems on manifolds arising from conformal geometry
Authors: Zhu, Huan (祝歡)
Abstract: This thesis focuses on the the fully nonlinear Yamabe-type problem on manifolds with
boundary of admissible negative curvature, which is an important problem in Geometric Analysis and has been extensively studied.
This problem is essentially an elliptic problem with Neumann boundary condition. Firstly, we use Maximum Principle and perturbation method to derive C0 bound.
Then, we creatively utilize the tubular neighborhood coordinates to derive C1 and C2
boundary estimates. When the C2 estimate is established, this problem turns out to be
uniformly elliptic. So, by the theory of Lieberman and Trudinger’s and the concave
condition, the C2,α estimate can be obtained. Furthermore, using the standard Schauder estimate, we can get C4,α estimate. Finally, the existence result is gained through
method of continuity and the uniqueness of this problem is derived by Maximum Principle.
Notes: CityU Call Number: QA649 .Z45 2012; iv, 83 p. 30 cm.; Thesis (Ph.D.)--City University of Hong Kong, 2012.; Includes bibliographical references (leaves [79]-83)Sun, 01 Jan 2012 00:00:00 GMThttp://dspace.cityu.edu.hk:80/handle/2031/72132012-01-01T00:00:00ZSome existence and stability problems of the Boltzmann equations
http://dspace.cityu.edu.hk:80/handle/2031/7204
Title: Some existence and stability problems of the Boltzmann equations
Authors: Wang, Ying (王穎)
Abstract: This thesis is concerned with some existence and stability theories on the Boltzmann
equations under certain conditions. The research for the Boltzmann equations
has been one of the most important and challenging field in Partial Differential
Equations because of its rich physical background and practical applications.
Thus, it is very important to reveal the properties of the Boltzmann equations
mathematically.
Fluid passing through porous media (e.g. the underground water passing
through the earth) can be modeled by the Euler equations with frictional force
which have been extensively studied. Since the Boltzmann equations are closely
related to the equations of gas dynamics, we investigate in the first part of this
thesis, the Boltzmann equation with frictional force when the external force is
proportional to the macroscopic velocity. We discuss the Cauchy problem of
the Boltzmann equations with frictional force mainly for the hard sphere model.
We give not only the existence theory but also the optimal time convergence
rates of the solutions to the Boltzmann equations with frictional force towards
equilibrium.
In the second part, we consider the specular re
flective boundary problem for
the one-dimensional Boltzmann equations with soft potentials. It is shown that
the solution converges to a global Maxwellian under certain initial conditions.
Note that the result for hard potentials case has already been established, thus
our result here is a good supplement of this problem.
Notes: CityU Call Number: QC173.4.P67 W37 2012; v, 74 leaves 30 cm.; Thesis (Ph.D.)--City University of Hong Kong, 2012.; Includes bibliographical references (leaves [71]-74)Sun, 01 Jan 2012 00:00:00 GMThttp://dspace.cityu.edu.hk:80/handle/2031/72042012-01-01T00:00:00Z