Relative controllability of linear stationary completely integrable Pfaff systems

We consider a process described by stationary completely integrable Pfaff systems linear in the input and the control as well as in the output and the state. We study the state controllability properties of the Pfaff system for the case in which an arbitrary initial state (a constant vector) and an arbitrary terminal state (either a constant vector, or an analytic vector function, or a constant vector and an analytic vector function) are given. The terminal states belong to a subspace of the state space of the system. The entire set of Pfaff systems splits into three classes, in each of which we obtain conditions for the controllability of Pfaff systems in the class of analytic controls. The conditions are stated in terms of the rank of some matrices formed from the given matrices of the Pfaff system and the matrix that specifies the subspace of terminal states.     

Numerical Taylor expansions of invariant manifolds in large dynamical systems

Summary. In this paper we develop a numerical method for computing higher order local approximations of invariant manifolds, such as stable, unstable or center manifolds near steady states of a dynamical system. The underlying system is assumed to be large in the sense that a large sparse Jacobian at the equilibrium occurs, for which only a linear (black box) solver and a low dimensional invariant subspace is available, but for which methods like the QR–Algorithm are considered to be too expensive. Our method is based on an analysis of the multilinear Sylvester equations for the higher derivatives which can be solved under certain nonresonance conditions. These conditions are weaker than the standard gap conditions on the spectrum which guarantee the existence of the invariant manifold. The final algorithm requires the solution of several large linear systems with a bordered Jacobian. To these systems we apply a block elimination method recently developed by Govaerts and Pryce [12, 14]. Received March 12, 1996 / Revised version reveiced August 8, 1997     

Existence of dichotomies and invariant splittings for linear differential systems,II

This paper is primarily concerned with linear time-varying ordinary differential equations. Sufficient conditions are given for the existence of an exponential dichotomy or equivalently an invariant splitting. The conditions are more general than those given in Part I of this paper and include the case in which the coefficients lie in a base space which is chain-recurrent under the translation flow and also the case in which compatible splittings are known to exist over invariant subsets of the base space. When the compatibility fails, the flow in the base space is shown to exhibit a gradient-like structure with attractors and repellers. Sufficient conditions are given guaranteeing the existence of bounded solutions of a linear system. The problem is treated in the unified setting of a skew-product dynamical system and the results apply to discrete systems including those generated by diffeomorphisms of manifolds. Sufficient conditions are given for a diffeomorphism to be an Anosov diffeomorphism.     

广义正交分解定理与Tseng度量广义逆

本文将Banach空间中广义正交分解定理从线性子空间拓广至非线性集—太阳集,分别给出了一算子为度量投影算子和一度量投影算子为有界线性算子的充要条件;得到了判别Banach空间中子空间广义正交可补的充要条件;建立了王玉文和季大琴(2000年)新近引入的Banach空间中的线性算子的Tseng度量广义逆存在的特征刻划条件;这些工作本质地把王玉文等人的新近结果从自反空间拓广至非自反空间的情形.     

Krein空间中无穷维Hamilton算子的极大确定不变子空间的存在性问题

本文研究了不定度规空间空间中的无穷维Hamilton算子.利用Plus算子存在极大不变子空间的性质,获得了无穷维Hamilton算子在Krein空间中存在极大确定不变子空间的充分条件.     

Some criteria for the existence of invariant measures and asymptotic stability for random dynamical systems on polish spaces

Tomasz Szarek presented interesting criteria for the existence of invariant measures and asymptotic stability of Markov operators on Polish spaces. Hans Crauel in his book presented the theory of random probabilistic measures on Polish spaces showing that notions of compactness and tightness for such measures are in one-to-one correspondence with such notions for non-random measures on Polish spaces, in addition to the criteria under which the space of random measures is itself a Polish space. This result allowed the transfer of results of Szarek to the case of random dynamical systems in the sense of Arnold. These criteria are interesting because they allow to use the existence of simple deterministic Lyapunov type function together with additional conditions to show the existence of invariant measures and asymptotic stability of random dynamical systems on general Polish spaces.     

Geometric Modular Action and Spontaneous Symmetry Breaking

We study spontaneous symmetry breaking for field algebras on Minkowski space in the presence of a condition of geometric modular action (CGMA) proposed earlier as a selection criterion for vacuum states on general space-times. We show that any internal symmetry group must commute with the representation of the Poincaré group (whose existence is assured by the CGMA) and each translation-invariant vector is also Poincaré invariant. The subspace of these vectors can be centrally decomposed into pure invariant states and the CGMA holds in the resulting sectors. As positivity of the energy is not assumed, similar results may be expected to hold for other space-times.Communicated by Klaus FredenhagenDedicated to the memory of Siegfried Schliedersubmitted 25/05/04, accepted 29/10/04     

An operator on a Fréchet space with no common invariant subspace with its inverse

A Fréchet space with a two-sided Schauder basis is constructed, such that the corresponding bilateral shift is continuous and invertible, and has no common nontrivial invariant subspace with its inverse. This shows in particular, that the problem of existence of hyperinvariant subspaces for operators on general Fréchet spaces, admits a negative answer. It is also shown that the dual of the Fréchet space constructed can be identified with a commutative locally convex complete topological algebra with unit, which has no closed nontrivial ideals.     

On invariant subspaces for polynomially bounded operators

We discuss the invariant subspace problem of polynomially bounded operators on a Banach space and obtain an invariant subspace theorem for polynomially bounded operators. At the same time, we state two open problems, which are relative propositions of this invariant subspace theorem. By means of the two relative propositions (if they are true), together with the result of this paper and the result of C.Ambrozie and V.Müller (2004) one can obtain an important conclusion that every polynomially bounded operator on a Banach space whose spectrum contains the unit circle has a nontrivial invariant closed subspace. This conclusion can generalize remarkably the famous result that every contraction on a Hilbert space whose spectrum contains the unit circle has a nontrivial invariant closed subspace (1988 and 1997).     

套子代数上线性映射的Lie不变子空间

本文引入了Banach代数上线性映射的Lie不变子空间,给出了因子VonNeumann代数中套子代数上以导子空间为Lie不变子空间的线性映射的一般形式,研究了Lie导子与Lie自同构的概念及了Lie导子与Lie自同构半群的关系.     

All-time Morse decompositions of linear nonautonomous dynamical systems

Morse decompositions provide inside information about the global asymptotic behavior of dynamical systems on compact metric spaces. Recently, the existence of Morse decompositions for nonautonomous dynamical systems was proved by restricting attention to the past or the future of the system, but in general, such a construction is not realizable for the entire time. In this article, it is shown that all-time Morse decompositions can be defined for linear systems on the projective space. Moreover, the dynamical properties are discussed and an analogue to the Theorem of Selgrade is proved.

     

The construction of Poincaré-Chetayevand Smale bifurcation diagrams for conservative non-holonomic systems with symmetry

The problem of the existence of first integrals which are linear functions of the generalized velocities (momenta and quasi-velocities) is discussed for conservative non-holonomic Chaplygin systems with symmetry, as well as methods for investigating the existence, stability, and bifurcation of the steady motions of such systems. These methods are based on the classical methods of Routh-Salvadori, Poincaré-Chetayev, and Smale, but unlike the latter they do not require a knowledge of the explicit form of the linear integrals. The general conclusions are illustrated by the example of the problem of an ellipsoid of revolution moving on an absolutely rough horizontal surface. It is shown how in this case numerical techniques can be used to construct the Poincaré-Chetayev diagram — a surface in the space of generalized coordinates and constants of linear first integrals corresponding to motions in which the velocities of the non-cyclic coordinates vanish, while those of the cyclic coordinates are constant, and the Smale diagram — a surface in the space of constants of linear first integrals and the energy integral corresponding to these motions.     

On invariant maximal non-negative subspaces of certain operators acting in a Krein space

The aim of this note is to show that a series of proofs for the existence of a maximal non-negative subspace which is invariant under an operator S in a Krein space, or for statements equivalent with this, follows a general pattern, using an approximating net S⁽ⁱ⁾ for S such that for S⁽ⁱ⁾ the existence of such a space is known.     

Invariant Measure for Diffusions with Jumps

Our purpose is to study an ergodic linear equation associated to diffusion processes with jumps in the whole space. This integro-differential equation plays a fundamental role in ergodic control problems of second order Markov processes. The key result is to prove the existence and uniqueness of an invariant density function for a jump diffusion, whose lower order coefficients are only Borel measurable. Based on this invariant probability, existence and uniqueness (up to an additive constant) of solutions to the ergodic linear equation are established. Accepted 24 February 1998     

一类混杂动态系统的能控性(Ⅰ)——基本结果

首次将时滞现象引入到线性切换系统的模型中,研究含有时滞线性切换系统的能控性及其判定条件。全部工作由三部分组成,第Ⅰ部分首先,提出含时滞的线性切换系统的数学模型,并介绍切换系统的基本概念—切换序列。其次,引入列空间、循环不变子空间和广义循环不变子空间等基本几何概念,给出一些有关概念的基本性质,特别是分离引理。然后以一个基本引理的形式揭式某一积分方程的解集与广义循环不变子空间之间的联系,这个引理将在能控性的判定中起关键作用。这些概念和引理都将作为以后展开能控性分析所必需的研究工具。     

On projective equivalence of invariant subspace lattices

For i = 1,2, let A_i be a linear transformation on a complex vector space and let σ be a lattice isomorphism from the invariant subspace lattice of A₁ onto the invariant subspace lattice of A₂. We determine the conditions under which σ is implemented by a linear or conjugate linear transformation (or a sum of these two kinds).     

The invariant subspace problem

A fundamental problem is to determine whether every bounded linear transformation in Hilbert space has a nontrivial invariant subspace. A formal proof [1] of the existence of invariant subspaces is given by the theory of square summable power series [2] in its vector formulation [3]. A determination of extreme points of a convex set remains for the justification of the formal argument. A characterization of extreme points which implies the existence of invariant subspaces has been conjectured [4]. New information is obtained from a localization of the theory of square summable power series [5] which allows the formulation of extreme point problems which are closely related because of the Carathéodory-Fejér extension theorem [6]. The conjectured characterization of extreme points is shown to be false. Extreme points need not have the properties required for the construction of invariant subspaces.     

The numerical approximation of center manifolds in Hamiltonian systems

In this paper we develop a numerical method for computing higher order local approximations of center manifolds near steady states in Hamiltonian systems. The underlying system is assumed to be large in the sense that a large sparse Jacobian at the equilibrium occurs, for which only a linear solver and a low-dimensional invariant subspace is available. Our method combines this restriction from linear algebra with the requirement that the center manifold is parametrized by a symplectic mapping and that the reduced equation preserves the Hamiltonian form. Our approach can be considered as a special adaptation of a general method from Numer. Math. 80 (1998) 1-38 to the Hamiltonian case such that approximations of the reduced Hamiltonian are obtained simultaneously. As an application we treat a finite difference system for an elliptic problem on an infinite strip.     

Weak global controllability of nonlinear systems

The problem of the controllability of nonlinear systems in finite-dimensional state spaces is considered in a measure-theoretical framework, in which we deal with a set of measured defined by the boundary conditions and the differential equations of the problem. The property of weak controllability is then equivalent to the existence of a positive measure satisfying a set of linear equalities and inequalities. This problem is solved by considering the extension of an associated linear functional, defined in a suitable subspace of the space of functions being used. Necessary and sufficient conditions for weak controllability are obtained.     

Internal gravity wave beams as invariant solutions of Boussinesq equations in geophysical fluid dynamics

It is shown that Lie group analysis of differential equations provides the exact solutions of two-dimensional stratified rotating Boussinesq equations which are a basic model in geophysical fluid dynamics. The exact solutions are obtained as group invariant solutions corresponding to the translation and dilation generators of the group of transformations admitted by the equations. The comparison with the previous analytic studies and experimental observations confirms that the anisotropic nature of the wave motion allows to associate these invariant solutions with uni-directional internal wave beams propagating through the medium. It is also shown that the direction of internal wave beam propagation is in the transverse direction to one of the invariants which corresponds to a linear combination of the translation symmetries. Furthermore, the amplitudes of a linear superposition of wave-like invariant solutions forming the internal gravity wave beams are arbitrary functions of that invariant. Analytic examples of the latitude-dependent invariant solutions associated with internal gravity wave beams that have different general profiles along the obtained invariant and propagating in the transverse direction are considered. The behavior of the invariant solutions near the critical latitude is illustrated.