5,446 research outputs found
Time-varying Bang-bang Property of Minimal Controls for Approximately Null-controllable Heat Equations
In this paper, optimal time control problems and optimal target control
problems are studied for the approximately null-controllable heat equations.
Compared with the existed results on these problems, the boundary of control
variables are not constants but time varying functions. The time-varying
bang-bang property for optimal time control problem, and an equivalence theorem
for optimal control problem and optimal target problem are obtained.Comment: 13 page
Observability inequalities from measurable sets for some evolution equations
In this paper, we build up two observability inequalities from measurable
sets in time for some evolution equations in Hilbert spaces from two different
settings. The equation reads: , and the observation operator is
denoted by . In the first setting, we assume that generates an analytic
semigroup, is an admissible observation operator for this semigroup (cf.
\cite{TG}), and the pair verifies some observability inequality from
time intervals. With the help of the propagation estimate of analytic functions
(cf. \cite{V}) and a telescoping series method provided in the current paper,
we establish an observability inequality from measurable sets in time. In the
second setting, we suppose that generates a semigroup, is a
linear and bounded operator, and the pair verifies some spectral-like
condition. With the aid of methods developed in \cite{AEWZ} and \cite{PW2}
respectively, we first obtain an interpolation inequality at one time, and then
derive an observability inequality from measurable sets in time. These two
observability inequalities are applied to get the bang-bang property for some
time optimal control problems.Comment: 29 page
Two equivalence theorems of different kinds of optimal control problems for Schr\"{o}dinger equations
This paper builds up two equivalence theorems for different kinds of optimal
control problems of internally controlled Schr\"{o}dinger equations. The first
one concerns with the equivalence of the minimal norm and the minimal time
control problems. (The minimal time control problems are also called the first
type of optimal time control problems.) The targets of the aforementioned two
kinds of problems are the origin of the state space. The second one deals with
the equivalence of three optimal control problems which are optimal target
control problems, optimal norm control problems and the second type of optimal
time control problems. These two theorems were estabilished for heat equations
in [18] and [17] respectively.Comment: 27 page
Approximation of time optimal controls for heat equations with perturbations in the system potential
In this paper, we study a certain approximation property for a time optimal
control problem of the heat equation with -potential. We prove that
the optimal time and the optimal control to the same time optimal control
problem for the heat equation, where the potential has a small perturbation,
are close to those for the original problem. We also verify that for the heat
equation with a small perturbation in the potential, one can construct a new
time optimal control problem, which has the same target as that of the original
problem, but has a different control constraint bound from that of the original
problem, such that the new and the original problems share the same optimal
time, and meanwhile the optimal control of the new problem is close to that of
the original one. The main idea to approach such approximation is an
appropriate use of an equivalence theorem of minimal norm and minimal time
control problems for the heat equations under consideration. This theorem was
first established by G.Wang and E. Zuazua in [20] for the case where the
controlled system is an internally controlled heat equation without the
potential and the target is the origin of the state space.Comment: 28 page
Minimal time control of exact synchronization for parabolic systems
This paper studies a kind of minimal time control problems related to the
exact synchronization for a controlled linear system of parabolic equations.
Each problem depends on two parameters: the bound of controls and the initial
state. The purpose of such a problem is to find a control (from a constraint
set) synchronizing components of the corresponding solution vector for the
controlled system in the shortest time. In this paper, we build up a necessary
and sufficient condition for the optimal time and the optimal control; we also
obtain how the existence of optimal controls depends on the above mentioned two
parameters
Group-theoretical analysis of variable coefficient nonlinear telegraph equations
Given a class of differential equations with arbitrary element, the problems
of symmetry group, nonclassical symmetry and conservation law classifications
are to determine for each member the structure of its Lie symmetry group,
conditional symmetry and conservation law under some proper equivalence
transformations groups. In this paper, an extensive investigation of these
three aspects is carried out for the class of variable coefficient
(1+1)-dimensional nonlinear telegraph equations with coefficients depending on
the space variable. The usual equivalence group and the extended one including
transformations which are nonlocal with respect to arbitrary elements are first
constructed. Then using the technique of variable gauges of arbitrary elements
under equivalence transformations, we restrict ourselves to the symmetry group
classifications for the equations with two different gauges g=1 and g=h. In
order to get the ultimate classification, the method of furcate split is also
used and consequently a number of new interesting nonlinear invariant models
which have non-trivial invariance algebra are obtained. As an application,
exact solutions for some equations which are singled out from the
classification results are constructed by the classical Lie reduction. The
classification of nonclassical symmetries for the classes of differential
equations with gauge g=1 is discussed within the framework of singular
reduction operator. Using the direct method, we also carry out two
classifications of local conservation laws up to equivalence relations
generated by both usual and extended equivalence groups. Equivalence with
respect to these groups and correct choice of gauge coefficients of equations
play the major role for simple and clear formulation of the final results.Comment: arXiv admin note: substantial text overlap with arXiv:0808.3577 by
other author
Equivalent Conditions on Periodic Feedback Stabilization for Linear Periodic Evolution Equations
This paper studies the periodic feedback stabilization for a class of linear
-periodic evolution equations.Several equivalent conditions on the linear
periodic feedback stabilization are obtained. These conditions are related with
the following subjects: the attainable subspace of the controlled evolution
equation under consideration; the unstable subspace (of the evolution equation
with the null control) provided by the Kato projection; the Poincar
map associated with the evolution equation with the null control; and two
unique continuation properties for the dual equations on different time
horizons and (where is the sum of algebraic
multiplicities of distinct unstable eigenvalues of the Poincar map).
It is also proved that a -periodic controlled evolution equation is linear
-periodic feedback sabilizable if and only if it is linear -periodic
feedback sabilizable with respect to a finite dimensional subspace. Some
applications to heat equations with time-periodic potentials are presented.Comment: 40 page
Attainable subspaces and the bang-bang property of time optimal controls for heat equations
In this paper, we study two subjects on internally controlled heat equations
with time varying potentials: the attainable subspaces and the bang-bang
property for some time optimal control problems. We present some equivalent
characterizations on the attainable subspaces, and provide a sufficient
conditions to ensure the bang-bang property. Both the above-mentioned
characterizations and the sufficient condition are closely related to some
function spaces consisting of some solutions to the adjoint equations. It seems
for us that the existing ways to derive the bang-bang property for heat
equations with time-invariant potentials (see, for instance, [4],[7],[16],[26])
do not work for the case where the potentials are time-varying. We provide
another way to approach it in the current paper.Comment: 33page
Quantitative unique continuation for the heat equation with Coulomb potentials
In this paper, we establish a H\"older-type quantitative estimate of unique
continuation for solutions to the heat equation with Coulomb potentials in
either a bounded convex domain or a -smooth bounded domain. The approach
is based on the frequency function method, as well as some parabolic-type Hardy
inequalities
Asymptotic Derivation and Numerical Investigation of Time-Dependent Simplified Pn Equations
The steady-state simplified Pn (SPn) approximations to the linear Boltzmann
equation have been proven to be asymptotically higher-order corrections to the
diffusion equation in certain physical systems. In this paper, we present an
asymptotic analysis for the time-dependent simplified Pn equations up to n = 3.
Additionally, SPn equations of arbitrary order are derived in an ad hoc way.
The resulting SPn equations are hyperbolic and differ from those investigated
in a previous work by some of the authors. In two space dimensions, numerical
calculations for the Pn and SPn equations are performed. We simulate neutron
distributions of a moving rod and present results for a benchmark problem,
known as the checkerboard problem. The SPn equations are demonstrated to yield
significantly more accurate results than diffusion approximations. In addition,
for sufficiently low values of n, they are shown to be more efficient than Pn
models of comparable cost.Comment: 32 pages, 7 figure
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