The Relation and Evolution of Squeezing and Instability for Systems with Quadratic Hamiltonians

We propose a method that allows relating the quantum squeezing effect to the classical instability by establishing evolution equations for elements of the dispersion matrix directly in terms of elements of the stability matrix. The solution of these equations is written in terms of the evolution operator. Knowing this operator, we can analyze the system instability at finite times. Based on the developed formalism, we investigate two physical systems: the degenerate and nondegenerate parametric amplifiers with external -shaped pulses. We show that we can either amplify or, on the contrary, weaken both the squeezing effect and the system instability using -pulses.     

Stability results for Hamilton-Jacobi equations with integro-differential terms and discontinuous Hamiltonians

In this paper we prove a stability result for Hamilton-Jacobi equations with an integro-differential term for discontinuous Hamiltonians. This type of equations arises in various problems concerning, for example, the control of diffusion processes with jumps, the theory of large deviations for processes with jumps, and the theory of piecewise deterministic processes.     

Stability and Instability of Schwarzschild-AdS for the Nonlinear Einstein-Klein-Gordon System

Acta Mathematicae Applicatae Sinica, English Series - In this paper, we study the global behavior of solutions to the spherically symmetric(it means that the problem is 2+2 framework)coupled...     

A Near-Linear Algorithm for the Planar 2-Center Problem

We present an -time algorithm for computing the 2-center of a set S of n points in the plane (that is, a pair of congruent disks of smallest radius whose union covers S), improving the previous -time algorithm of [10]. Received May 2, 1995, and in revised form July {8, 1995}.     

Stability and Instability of Steady States for a Branching Random Walk

Methodology and Computing in Applied Probability - We consider the time evolution of a lattice branching random walk with local perturbations. Under certain conditions, we prove the Carleman type...     

Dynamic Stability and Instability of Pinned Fundamental Vortices

We study the dynamic stability and instability of pinned fundamental ±1 vortex solutions to the Ginzburg–Landau equations with external potential in ℝ². For sufficiently small external potentials, there exists a perturbed vortex solution centered near each non-degenerate critical point of the potential. With respect to both dissipative and Hamiltonian dynamics, we show that perturbed vortex solutions which are concentrated near local maxima (resp. minima) are orbitally stable (resp. unstable). In the dissipative case, the stability is in the stronger “asymptotic” sense. The research of S. Gustafson partially supported by a grant from NSERC. The research of F. Ting on this paper was supported by NSERC under grant N298724.     

Optimal Stability and Eigenvalue Multiplicity

Abstract. We consider the problem of minimizing over an affine set of square matrices the maximum of the real parts of the eigenvalues. Such problems are prototypical in robust control and stability analysis. Under nondegeneracy conditions, we show that the multiplicities of the active eigenvalues at a critical matrix remain unchanged under small perturbations of the problem. Furthermore, each distinct active eigenvalue corresponds to a single Jordan block. This behavior is crucial for optimality conditions and numerical methods. Our techniques blend nonsmooth optimization and matrix analysis.     

The Strong Stability and Instability of a Fluid-Structure Semigroup

The strong stability problem for a fluid-structure interactive partial differential equation (PDE) is considered. The PDE comprises a coupling of the linearized Stokes equations to the classical system of elasticity, with the coupling occurring on the boundary interface between the fluid and solid media. Because of the nature of the unbounded coupling between fluid and structure, the resolvent of the associated semigroup generator will not be a compact operator. In consequence, the classical solution to the stability problem, by means of the Nagy-Foias decomposition, will not avail here. Moreover, it is not practicable to write down explicitly the resolvent of the fluid-structure generator; this situation thus makes it problematic to use the well-known semigroup stability result of Arendt-Batty and Lyubich-Phong. When a locally supported boundary dissipative mechanism is in place, we derive here a result of strong decay for this fluid-structure PDE. In the absence of said dissipative mechanism, we show the lack of asymptotic decay for solutions corresponding to arbitary initial data of finite energy.     

灰色离散系统稳定及不稳定的代数判据

本文通过作者提出的区间矩阵离散稳定性的概念,结合矩阵理论的方法,给出了灰色离散线性系统稳定性及不稳定性的一些实用代数判据.另外,本文还给出了区间矩阵稳定的几个简洁判据.最后给出了应用实例.     

Hamiltonians and conjugate Hamiltonians of some fourth-order nonlinear ODEs

We first derive the Lagrangians of the reduced fourth-order ordinary differential equations studied by Kudryashov under the assumption that they satisfy the conditions stated by Fels [M.E. Fels, The inverse problem of the calculus of variations for scalar fourth-order ordinary differential equations, Trans. Amer. Math. Soc. 348, 1996, 5007-5029], using Jacobi’s last multiplier technique. In addition we derive the Hamiltonians of these equations using the Jacobi-Ostrogradski theory. Next, we derive the conjugate Hamiltonian equations for such fourth-order equations passing the Painlevé test. Finally, we investigate the conjugate Hamiltonian formulation of certain additional equations belonging to this family.     

Asymptotic velocity for fibered Hamiltonians

We show the existence of the asymptotic velocity for a large class of Hamiltonians with a fibered structure.     

A Stability Theorem for Constrained Optimal Control Problems

This paper presents the stability of difference approximations of an optimal control problem for a quasilinear parabolic equation with controls in the coefficients, boundary conditions and additional restrictions. The optimal control problem has been convered to one of the optimization problem using a penalty function technique. The difference approximations problem for the considered problem is obtained. The estimations of stability of the solution of difference approximations problem are proved. The stability estimation of the solution of difference approximations problem by the controls is obtained.     

关于变系数线性方程的稳定性和不稳定性

本文研究变系数线性方程零解稳定性。给出了不稳定的充要条件，同时给出了一类系统渐近稳定的充要条件，还给出了判别稳定性的较为简易的充分条件。     

Optimal Input Design for Aircraft Stability and Control Flight Testing

Journal of Optimization Theory and Applications - Flight testing to characterize aircraft stability and control is a complex and expensive task that involves restrictive practical constraints....     

Semicontinuous viscosity solutions for quasiconvex Hamiltonians

The main theorem connecting convex Hamiltonians and semicontinuous viscosity solutions due to Barron and Jensen is extended to quasiconvex Hamiltonians. Some applications are indicated.     

Hyperoctahedral Chen calculus for effective Hamiltonians

We tackle the problem of unraveling the algebraic structure of computations of effective Hamiltonians. This is an important subject in view of applications to chemistry, solid state physics or quantum field theory. We show, among other things, that the correct framework for these computations is provided by the hyperoctahedral group algebras. We define several structures on these algebras and give various applications. For example, we show that the adiabatic evolution operator (in the time-dependent interaction representation of an effective Hamiltonian) can be written naturally as a Picard-type series and has a natural exponential expansion.