New results on the parametric stability of nonlinear systems

In this paper, we derive some new results on the parametric stability of nonlinear systems. Explicitly, we derive a necessary and sufficient condition for a nonlinear system to be locally parametrically exponentially stable at an equilibrium point. We also derive a necessary condition for the nonlinear system to be locally parametrically asymptotically stable at an equilibrium point. Next, we derive some new results on the parametric stability of discrete-time nonlinear systems. As in the continuous case, we derive a necessary and sufficient condition for a discrete-time nonlinear system to be locally parametrically exponentially stable at an equilibrium point. We also derive a necessary condition for the discrete-time nonlinear system to be locally parametrically asymptotically stable at an equilibrium point. We illustrate our results with some classical examples from the bifurcation theory.     

CHARACTERISTIC MATRICES AND SIMILARITY OF TIME-VARYING LINEAR DIFFERENTIAL SYSTEMS

1IntroductionLineardial'~rentialsystemsa.rethesimplestmod(}lsinmotiolls,tijoyarcwidelyusedincontroltheory,Inechanlcsandengineel'illg.Invest;igationsoiltileasymptoticpropertiesoftinea,rSystclllshay')oss(3ntialsense.Fbi'tilelin(3arsy'stemswithconstantcoeffi…     

The Basis Properties of Some Systems of Exponential Functions,Cosines, and Sines

We consider some systems of exponential functions, cosines, and sines with complex-valued coefficients and establish a necessary and sufficient condition for completeness and minimality of these systems in Lebesgue spaces.     

A necessary and sufficient convergence condition of orthomin(k) methods for least squares problem with weight

A class of Orthomin-type methods for linear systems based on conjugate residuals is extended to a form suitable for solving a least squares problem with weight. In these algorithms a mapping matrix as preconditioner is brought into use. We also give a necessary and sufficient condition for the convergence of the algorithm. Furthermore, we also study the construction of the mapping matrix for which the necessary and sufficient condition holds.     

Conditions for the solvability of boundary value problems for quasi-elliptic systems in a half-space

We consider boundary value problems in a half-space for a class of quasi-elliptic systems with constant coefficients. We assume that the boundary value problems satisfy the Lopatinskii condition. We obtain necessary and sufficient conditions for their unique solvability in Sobolev spaces.     

Integrability of Hamiltonian systems and the Lamé equation

We study the integrability of Hamiltonian systems with two degrees of freedom. We investigate the normal variational equations and obtain a necessary condition for integrability of these systems. As an application we study the integrability of the Hénon–Heiles system, whose normal variational equation is of Lamé type.     

相互粒子系统中的负相关(英文)

主要研究相互粒子系统中概率测度的负相关.我们得到判定概率测度是负相关的一个充分必要条件.最后证明了具有负相关的概率测度的线性组合及乘积测度仍是负相关的.     

A hypergroupal characterization of projective planes

We give two sufficient conditions for a hypergroupoid to be a feeble semi-hypergroupoid and a sufficient condition to be a feeble hypergroup. We show that every hypergroup in the class of the k-quasi-Steiner hypergroupoids is feebly associative. Finally, we apply these algebraic results to study Steiner systems; in particular, we give a necessary and sufficient condition for a Steiner system to be a projective plane.     

时变系统的稳定化

This paper deals with the stabilization problem for linear time-varying systems within the framework of nest algebras. We give a necessary and sufficient condition for a class of plants to be stabilizable, and we also study the simultaneous and strong stabilization problems.     

线性离散系统的部分稳定性

In this paper, we study the partial stability of linear discrete systems by means of Liapunov's functions of quadratic form . We obtain a necessary and sufficient condition for the system being stable with respect to part of variables and generalize Liapunov's equation to the partial stability of linear discrete systems . A method of constructing Liapunov's function of quadratic form for the stability of the systems is given.     

The problem of coordination and consensus achievement in groups of autonomous mobile robots with limited communication

The paper considers a consensus seeking problem for a multirobot system with limited communication. Multirobot systems with limited communication are modelled as hybrid dynamical systems. The proposed model is a modification of some mathematical models introduced in physics and biology. We derive a necessary and sufficient condition for the system to converge to a single value of the coordination variable with probability 1.     

Correct solvability of Solonnikov–Eidel’man parabolic initial-value problems

We consider initial-value problems for a new class of systems of equations that combine the structures of Solonnikov parabolic systems and Eidel’man parabolic systems. We prove a theorem on the correct solvability of these problems in H?lder spaces of rapidly increasing functions and obtain an estimate for the norms of solutions via the corresponding norms of the right-hand sides of the problem. For the correctness of this estimate, the condition of the parabolicity of the system is not only sufficient but also necessary.     

二环链系统的稳定性和能控性

提出了二环链系统,并利用二环链特征向量,给出了一个系统可实现二环链分解的充分必要条件,给出了可实现二环链线性系统的分解算法,研究了二环链系统的稳定性和能控性.     

Consistent systems and pole assignment: I

We study the consistency property of a linear time-invariant control system with incomplete feedback and of a bilinear time-invariant control system in connection with the pole assignment problem. We show that consistency is a sufficient and, in some cases, necessary condition for the arbitrary pole assignability of systems with coefficients of a special form.     

2-Regularity and 2-Normality Conditions in Discrete Optimal Control Problems

The paper is devoted to general optimal control problems for discrete-time systems with equality type of constraints on end points and control. We derive first-order necessary optimality conditions that are meaningful under the new nontriviality condition.     

Binding of atoms and stability of molecules in hartree and thomas—fermi type theories

This paper is the sequel of a previous work where we showed a general necessary and sufficient condition for the stability of an arbitrary molecular system (possibly ionized) in the framework of Hartree or Thomas-Fermi type theories. This condition, roughly speaking, meant that certain particular subsystems have to be bound. We show here in particular that this condition reduces for general molecular system with nonnegative excess charge to the binding of all subsystems with the same property. For neutral inolecular systems, this reduces to the binding of all neutral subsystems. In both cases, all other subsystems can be bound. We also show that, for the Hartree-Fock and Hartree models, this condition involves only “physical” sulxystems We use these reduced conditions to conclude allout the stability or the binding in some particular cases. This work 1s also the second of a series devoted to these equations and we shall come back on the binding of neutral systems in Part 3.     

On the Convergence of Formal Solutions of a System of Partial Differential Equations

We study a version of the classical problem on the convergence of formal solutions of systems of partial differential equations. A necessary and sufficient condition for the convergence of a given formal solution (found by any method) is proved. This convergence criterion applies to systems of partial differential equations (possibly, nonlinear) solved for the highest-order derivatives or, which is most important, “almost solved for the highest-order derivatives.”     

关于线性方程组同解的条件

讨论线性方程组同解的条件,得到了两个线性方程组同解的充分必要条件.     

The sequential differentiation and its applications in the optimal control problems

Using the sequential approach, we define a certain generalization of the operator derivative. We establish the necessary extremum condition in terms of the sequential derivative. As examples we consider the optimal control problems for systems governed by partial nonlinear differential equations of several kinds.     

Euler-Lagrange and Hamiltonian formalisms in dynamic optimization

We consider dynamic optimization problems for systems governed by differential inclusions. The main focus is on the structure of and interrelations between necessary optimality conditions stated in terms of Euler-Lagrange and Hamiltonian formalisms. The principal new results are: an extension of the recently discovered form of the Euler-Weierstrass condition to nonconvex valued differential inclusions, and a new Hamiltonian condition for convex valued inclusions. In both cases additional attention was given to weakening Lipschitz type requirements on the set-valued mapping. The central role of the Euler type condition is emphasized by showing that both the new Hamiltonian condition and the most general form of the Pontriagin maximum principle for equality constrained control systems are consequences of the Euler-Weierstrass condition. An example is given demonstrating that the new Hamiltonian condition is strictly stronger than the previously known one.