One-sided Boundary Research Articles

Threshold optimization in observability inequality for the wave equation with homogeneous Robin-type boundary condition

For the wave equation with variable coefficients, problems with one-side boundary controls of three basic types and a boundary condition of the third kind at the uncontrolled end are considered. For dual problems with one-side boundary observations in the classes of strong generalized solutions, new constructive observability inequalities are obtained that are superior to the earlier known ones in two respects. First, inequalities with an optimal value of the controllability-observability threshold are derived, and second, the value of the final evaluation constant is bounded away from zero on time intervals whose length is close to the critical length. This opens up a possibility of constructing stable approximate solutions to the indicated classes of dual control and observation problems on time intervals not only of an arbitrary supercritical but also of precisely critical length.

New construction of the magnetohydrodynamic spectrum of stationary plasma flows. I. Solution path and alternator

A new method of systematically constructing the full structure of the complex magnetohydrodynamic spectra of stationary flows is presented. It is based on the self-adjointness of the generalized force operator G and the Doppler–Coriolis shift operator U, and the associated quadratic forms for the normalized energy W¯ and the normalized Doppler–Coriolis shift V¯, which may be constructed for all complex values of ω if the original eigenvalue problem is converted into a one-sided boundary value problem. This turns W¯ into a complex expression, while V¯ remains real. Whereas the solution path Ps of stable modes is just the real axis, the solution path Pu of unstable modes in the complex ω plane is found by requiring that the solution-averaged Doppler–Coriolis shifted real part of the frequency vanishes, σ−V¯[ξ(ω)]=0, or that the energy is real, Im W¯[ξ(ω)]=0. The location of the eigenvalues on these solution paths is determined by two quadratic forms, which may straightforwardly be evaluated in any of the finite element spectral codes in existence. A new oscillation theorem is proved about the monotonicity of complex eigenvalues for one-dimensional systems. Instead of counting internal nodes of the real displacement vector ξ (as in static plasmas), it is based on counting the zeros of the alternating ratio, or alternator, R≡ξ/Π of the boundary values of the complex functions ξ and the total pressure perturbation Π, which is real on the solution path. This finally provides the generalization of the basic structural properties of the magnetohydrodynamic spectrum of static plasmas, which has been known for a long time, to stationary plasmas.

Steady Euler–Poisson systems: A differential/integral equation formulation with general constitutive relations

The Cauchy problem and the initial-boundary value problem for the Euler–Poisson system have been extensively investigated, together with a study of scaled and unscaled asymptotic limits. The pressure–density relationships employed have included both the adiabatic (isentropic) relation as well as the ideal gas law (isothermal). The study most closely connected to this one is that of Nishibata and Suzuki [S. Nishibata and M. Suzuki, Asymptotic stability of a stationary solution to a hydrodynamic model of semiconductors, Osaka J. Math. 44 (2007) 639–665], where a power law was employed in the context of the subsonic case, covering both the isothermal and adiabatic cases. These authors characterize the steady solution as an asymptotic limit. In this paper, we consider only the steady case, in much greater generality, and with more transparent arguments, than heretofore. We are able to identify both subsonic and supersonic regimes, and correlate them to one-sided boundary values of the momentum and concentration. We employ the novelty of a differential/integral equation formulation.

Exact boundary controllability for a kind of second-order quasilinear hyperbolic systems and its applications

In this paper, by means of a constructive method based on the existence and uniqueness of the semi-global C2 solution, we establish the local exact boundary controllability for a kind of second-order quasilinear hyperbolic systems. As an application, we obtain the one-sided local exact boundary controllability for the first-order quasilinear hyperbolic systems of diagonal form with boundary conditions in which the diagonal variables corresponding to the positive eigenvalues and those corresponding to the negative eigenvalues are decoupled. Copyright © 2009 John Wiley & Sons, Ltd.

A note on the one-side exact boundary controllability for quasilinear hyperbolic systems

The known theory on the one-side exact boundary observabil- ity for first order quasilinear hyperbolic systems requires that the unknown variables are suitably coupled or satisfy the Group Property in boundary conditions on the non-observation side (see (1)-(2), (11)). In this paper we illustrate, with an inspiring example, that the one-side exact boundary ob- servability can be realized by means of a suitable coupling of the unknown variables in quasilinear hyperbolic system itself instead of in boundary con- ditions. Moreover, an implicit duality between the one-side exact boundary controllability and one-side exact boundary observability is also revealed in this situation.

First Passage Densities and Boundary Crossing Probabilities for Diffusion Processes

We consider the boundary crossing problem for time-homogeneous diffusions and general curvilinear boundaries. Bounds are derived for the approximation error of the one-sided (upper) boundary crossing probability when replacing the original boundary by a different one. In doing so we establish the existence of the first-passage time density and provide an upper bound for this function. In the case of processes with diffusion interval equal to ℝ this is extended to a lower bound, as well as bounds for the first crossing time of a lower boundary. An extension to some time-inhomogeneous diffusions is given. These results are illustrated by numerical examples.

Riser-Soil Interaction: Local Dynamics at TDP and a Discussion on the Eigenvalue and the VIV Problems

Abstract The eigenvalue problem of risers is of utmost importance, particularly if vortex-induced vibration (VIV) is concerned. Design procedures rely on the determination of eigenvalues and eigenmodes. Natural frequencies are not too sensitive to the proper consideration of boundary condition, within a certain extent where dynamics at the touchdown area (TDA) may be modeled as dominated by the dynamics of the suspended part. However, eigenmodes may be strongly affected in this region because, strictly speaking, this is a nonlinear one-side (contact-type) boundary condition. Actually, we shall consider a nonlinear eigenvalue problem. Locally, at TDA, riser flexural rigidity and soil interaction play important roles and may affect the dynamic curvature. Extending and merging former analytical solutions on touchdown point (TDP) dynamics and on the eigenvalue problem, obtained through asymptotic and perturbation methods, the present work critically address soil and bending stiffness effects a little further. As far as linear soil stiffness and planar dynamics hypotheses may be considered valid, it is shown that penetration in the soil is small and that its effect does not change significantly the bending loading that is mainly caused by the cyclic excursion of the TDP and corresponding dynamic tension. A comparison of the analytical results with a full nonlinear time-domain simulation shows a remarkable agreement for a typical steel catenary riser. The WKB approximation for the eigenvalue problem gives good estimates for TDP excursion. As the dynamic tension caused solely by VIV is very small, the merged analytical solution may be used as a first estimate of the curvature variation at TDP in the cases of current perpendicular to the “riser plane.”

Existence of solutions of parabolic variational inequalities with one-sided restrictions

We prove sufficient conditions for the existence of a solution of a “strong” nonlinear variational inequality of parabolic type. The theory can be used for solving parabolic equations with one-sided boundary conditions. As an example, we prove the existence of a solution of a “strong” parabolic variational inequality with p-Laplacian in the Sobolev space L p (0, T, W p 1 (Ω)), p ∈ [2, ∞).

Approximate Method of Solving the Problem of Diffraction of a Plane Electromagnetic Wave by a Thin Chiral Layer Covering a Perfectly Conducting Plane

One-sided approximate impedance-type boundary conditions for a thin chiral layer placed on a perfectly conducting plane are derived. With these conditions, the problem of incidence of a plane electromagnetic wave on a chiral structure is solved. Approximate formulas for the coefficients of reflection of the fundamental and depolarized components are derived for the case of the perpendicular polarization of the electromagnetic wave (the electric field strength is normal to the plane of incidence). A comparison with an exact solution to the problem of diffraction by this chiral structure is made.

On the First-Passage Time of a Diffusion Process Over a One-Sided Stochastic Boundary

Some problems about the asymptotics of the first-passage time of a one-dimensional diffusion process through a stochastic, as well as deterministic one-sided boundary are studied, generalizing the analogous results found by Peskir and Shiryaev for Brownian motion.

One-Sided Boundary Control for the Process of Oscillations Described by the Equation k ( x )[ k ( x ) u x ( x , t )] x − u tt ( x , t ) = 0

This article concerns the boundary control on the side x = 0 when the other side x = l is fixed. The corresponding process of oscillations is studied for the time interval T > 0 and is described by the generalized solution of the equation $$k\left( x \right)\left[ {k\left( x \right)u_x \left( {x,t} \right)} \right]_x u_{tt} \left( {x,t} \right) = 0$$ This solution belongs to the class that admits the existence of finite energy for any time moment t.

Asymptotics of First-Passage Time Over a One-Sided Stochastic Boundary

We study the asymptotic behavior of the first-passage times for Brownian motion, Levy processes and continuous martingales over one-sided increasing stochastic, as well as deterministic, boundaries. In particular, we study the first-passage time of a Brownian motion over the increasing function of its local time, give necessary and sufficient conditions for t−1/2 asymptotics, and obtain exact asymptotics for linear functions.

On the Brownian First-Passage Time Overa One-Sided Stochastic Boundary

Let $B=(B_t)_{t \g 0}$ be standard Brownian motion started at 0 under P, let $S_t=\max_{ 0 \l r \l t} B_r$ be the maximum process associated with B, and let $g: {\bf R}_+ \rightarrow {\bf R}$ be a (strictly) monotone continuous function satisfying $g(s) 0 \vert B_t \l g(S_t) \big \}. $$ Let G be the function defined by $$ G(y) = \exp \bigg(-\int_0^{g^{-1}(y)} {{ds} \over {s-g(s)}} \bigg) $$ for $y \in \bf R$ in the range of g. Then, if g is increasing, we have $$ \lim_{t \rightarrow \infty} \sqrt{t} \bP\{\tau \g t \} = \sqrt{2 \over \pi } \Bigg(-g(0) -\int_{g(0)}^{g(\infty)} G(y)\,dy \Bigg) $$ and this number is finite. Similarly, if g is decreasing, we have $$ \lim_{t \rightarrow \infty} \sqrt{t} \bP\{\tau \g t \} = \sqrt{2 \over \pi } \Bigg(-g(0) + \int_{g(\infty)}^{g(0)} G(y) \,dy \Bigg) $$ and this number may be infinite. These results may be viewed as a {\it stochastic boundary} ext...

A heat flux limiter finite difference scheme in solving a one-dimensional inverse Stefan problem

This paper presents an enthalpy formulation of a finite difference scheme for solving an inverse Stefan problem in one space variable. Sometimes measurements cannot access the inside of the intended test domain; the information available may be taken only from one-side surface. The current inversion procedure considers one-side surface boundary conditions (i.e. both the surface temperature and the heat flux at one boundary) to recover the unknown interior temperature history and the melting front position. A semi-explicit time marching finite difference scheme with a heat flux limiter is used to handle the oscillations in the recovered temperature caused by the ill-posedness of the inverse problem, and to accelerate the rate of convergence of the iterations. Numerical recovered results show that solutions obtained with the heat flux limiter are much better than those obtained without it.

Boundary-controlled quantum transport in narrow wires

Transport properties of a quasi-ballistic wire with a point-contact (PC) in one-side boundary were studied. In the two-terminal resistance of the wire, when the width were sufficiently narrow, we have observed two competing sets of plateaus. They were able to be distinguished with each other by their different behaviours against variations of a sample size as well as a boundary condition. One of the plateau sets among them was found to originate from the PC itself in the wire boundary. This result indicates a possibility of a new class of narrow wires having a specific transport property due to the tailored quantum boundaries connecting with reservoirs or another wires.

Two-level system coupled to a boson mode.

The thermodynamics of a two-level system coupled to a boson field is investigated by means of a functional-integral method. The model can be used to describe the motion of a small polaron between two sites on a molecule, or a spin-1/2 magnetic moment coupled to a phonon bath. The model is also relevant to the problem of dissipation in macroscopic quantum phenomena. We use a functional-integral method to investigate a two-level electronic system coupled linearly to a harmonic oscillator. The density matrix is cast as a functional integral over paths obeying one-sided boundary conditions at either 0 or \ensuremath{\beta}=1/${\mathit{k}}_{\mathit{B}}$T. The integration variables are c numbers for the oscillator and Grassmann numbers for the electrons. The oscillator variables are integrated out, leaving an effective density matrix for the electrons containing retarded interactions. The partition function is then calculated up to second order of an expansion about the exact fermionic classical paths. Then, the free energy, specific heat, and average hopping energy are obtained. We find that the most significant reduction in the hopping energy is manifested at intermediate temperatures.

One-Sided Boundary Crossing for Processes with Independent Increments

On etudie le comportement de la probabilite P(τ>t), t→∞, ou τ est le premier instant ou un processus X t franchit une frontiere f: τ=inf{t:X t >f(t)}

Generating functional for electron-phonon systems from functional integrals with one-sided boundary conditions.

We derive the generating functional of thermal Green's functions for an interacting electron-phonon system through the use of functional integrals in holomorphic representations (c numbers for phonons and Grassmann numbers for electrons). These integrals are over paths obeying one-sided boundary conditions at either 0 or \ensuremath{\beta}=1/kT, and do not require normalization factors unlike previous derivations. We recover the Matsubara perturbation formalism without using any results from the standard operator method.

Functional integral for the molecular polaron with q2 coupling.

The density matrix of a two-state electronic system coupled quadratically to an oscillator is expressed as functional integrals over commuting and anticommuting (Grassmann) elements satisfying one-sided boundary conditions. The functional integral over the oscillator coordinates is evaluated in closed form, and the resultant electronic integral which contains effective retarded interactions is evaluated in the semiclassical approximation. In the case of physical interest, N=1, we find that the semiclassical approximation already incorporates the essential physical features of the exact solution, for all values of the interaction strength and of the temperature.

On a property of lattice paths

It is well known that the number of lattice paths restricted by a one-sided boundary satisfies a recurrence relation involving coefficients which are counting numbers for unrestricted paths. In the same relation if the coefficients become counting numbers forrestricted paths it is shown that the number satisfying the new recurrence relation is a counting number for paths restricted by a new boundary.