最小多项式矩阵与线性多变量系统(Ⅱ)

本文之(Ⅰ)[8]是关于最小多项式矩阵的理论;其(Ⅱ)是关于这一理论在线性多变量系统中的应用.在本部分的第一节中,我们利用(Ⅰ)中的理论,详细地讨论线性多变量系统输入问题的一些结果.在第二节中,利用对偶性,我们给出行n.p.m.及行生成组等概念,并讨论线性多变量系统输出问题的某些结果.在第三节中.我们讨论化状态空间型为多项式矩阵型的方法.在第四节中,我们讨论这一问题的反问题,即化多项式矩阵型为状态空间型的问题.为说明这些理论和方法,我们给出一些有趣的例子.     

Comparison of decay of solutions to two compressible approximations to Navier-Stokes equations

In this article, we use the decay character of initial data to compare the energy decay rates of solutions to different compressible approximations to the Navier- Stokes equations.We show that the system having a nonlinear damping term has slower decay than its counterpart with an advection-like term. Moreover, me characterize a set of initial data for which the decay of the first system is driven by the difference between the full solution and the solution to the linear part, while for the second system the linear part provides the decay rate.     

Exponential mean square stability of partially linear stochastic systems

The purpose of this paper is to state sufficient conditions for the existence of linear feedback laws which render the equilibrium solution of a composite partially linear stochastic system (the linear part of which is deterministic) exponentially stable in mean square.     

Stability,estimation of state,and control in one nonlinear system

A controlled system of nonlinear ordinary differential equations is considered. In this system, we distinguish the time-varying linear part, the part that linearly depends on the control, and the nonlinearity, which vanishes on the steady-state solution and satisfies the Lipschitz condition. In addition, there are some constraints imposed on the control.     

Two-dimensional homogeneous cubic systems: Classification and normal forms. I

This work is the first in a series of papers devoted to classifying of two-dimensional homogeneous cubic systems based on partitioning into classes of linear equivalence. Principles have been developed that are capable of constructively distinguishing the structure of a simplest system in each class and a canonical set that defines the admissible values that can be assumed by its coefficients. The polynomial vector in the right-hand part of this system identified with a 2 × 4 matrix is called the canonical form (CF) and the system itself is called the normal cubic form. One of the main objectives of this series of papers is to maximally simplify the reduction of a system with a homogeneous cubic polynomial in the unperturbed part to the various structures of a generalized normal form (GNF). Generalized normal form refers to a system in which the perturbed part has the simplest form in some sense. The constructive implementation of the normalization process depends on the ability to explicitly specify the conditions of compatibility and possible solutions of the so-called bonding system, which is understood to be a countable set of linear algebraic equations that specify the normalizing transformations of the perturbed system. The above principles are based on the idea of the maximum possible simplification of the bonding system. This will allow one to first reduce the initial perturbed system by an invertible linear substitution of variables to a system with some CF in the unperturbed part, then reduce the resulting system, which is optimal for normalization, by almost identical substitutions to various structures of the GNF. In this paper, the following tasks are carried out: (1) the general problem is set, close problems are formulated, and the available results are described; (2) a bonding system is derived that is capable of determining the equivalence of any two perturbed systems with the same homogeneous cubic part, the possibilities of its simplification are discussed, the GNF is defined, and the method of resonant equations is given allowing one to constructively obtain all its structures; (3) special forms of recording homogeneous cubic systems in the presence of a common homogeneous factor in their right-hand parts with a degree of 1–3 are introduced, and the linear equivalence of these systems, as well as of systems without a common factor is studied, and key linear invariants are offered.     

Controllability and synthesis of control for nonlinear systems

We solve motion control problems with feedback under the linear approximation of equations of the considered system. Due to the smallness of the nonlinear part and the controllability of the linear approximating system this property is preserved under a nonlinear perturbation. We perform the study by the method of Lyapunov functions and by the comparison method within finite and infinite time intervals. We illustrate the obtained results by an example of controlling a simple mechanical system.     

On a system of reaction-diffusion equations describing a population with two age groups

A system of reaction-diffusion equations modelling a population divided into juvenile and adult age groups is studied. The system is not co-operative but its linear part is and this makes it possible to establish existence, non-existence and stability results for non-negative solutions of the system in terms of the principal eigenvalue of the corresponding linearized system.     

Multiobjective Optimization including Safety of Operation Applied to a Linear Drive System

In this contribution, we introduce a multiobjective optimization used to calculate safe optimal working points for a mechatronic system by including stochastic safety-critical signals in an objective function. Our application example consists of a linear drive for a rail-bound vehicle and an actuation unit. The linear drive's secondary part is fixed; the primary part is vehicle-mounted and can be adjusted vertically to account for deviations of the height of the secondary part. A small air gap between both parts improves efficiency, but increases the risk of a collision between the two parts. Using height data of the secondary part, a trajectory for the vertical adjustment of the primary part is calculated. However, unexpected deviations necessitate a readjustment of the air gap. The probability of such unexpected height deviations can be calculated from the readjustment data. The system is equipped with sensors to measure the air gap. Assuming that the sensor noise is normally distributed, noise characteristics are determined. Using this information and the probability distribution of unexpected height deviations, the probability of a collision is determined.T he sensor noise and the probability of a collision between both parts of the linear drive are included in the dynamical model of the system. Using multiobjective optimization, pareto-optimal working points for the controller of the air gap are obtained. By selecting an appropriate working point, safe operation can be ensured. (© 2013 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Additive Schwarz algorithms for parabolic convection-diffusion equations

Summary In this paper, we consider the solution of linear systems of algebraic equations that arise from parabolic finite element problems. We introduce three additive Schwarz type domain decomposition methods for general, not necessarily selfadjoint, linear, second order, parabolic partial differential equations and also study the convergence rates of these algorithms. The resulting preconditioned linear system of equations is solved by the generalized minimal residual method. Numerical results are also reported.This work was supported in part by the National Science Foundation under Grant NSF-CCR-8903003 at the Courant Institute, New York University and in part by the National Science Foundation under contract number DCR-8521451 and ECS-8957475 at Yale University     

Synthesis of the Optimal Control of Linear Logical-Dynamical Systems Under Instant Multiple Switchings of the Automaton Part

We consider a dynamic system controlled by an automaton with memory. The continuous part of the system is described by linear differential equations and the logical (automaton) part is described by linear recurrence equations. The moments of the state change of the automaton part are not known in advance and they are determined in the process of optimization. Modes with instant multiple switchings of the automaton part are admitted. Based on sufficient optimality conditions, we develop a technique for the synthesis of a feedback control. The application of the technique is demonstrated in an example.     

Approximate controllability of differential inclusions in Hilbert spaces

In this paper, controllability for the system originating from semilinear functional differential equations in Hilbert spaces is studied. We consider the problem of approximate controllability of semilinear differential inclusion assuming that semigroup, generated by the linear part of the inclusion, is compact and under the assumption that the corresponding linear system is approximately controllable. By using resolvent of controllability Gramian operator and fixed point theorem, sufficient conditions have been formulated and proved. An example is presented to illustrate the utility and applicability of the proposed method.     

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.     

A computer graphics-based Decision Support System for multiple objective linear programming

In this paper we propose a computer-graphics based Decision Support System for multiple objective linear programming that builds on the VIG system (Visual Interactive Goal programming). The essential part of the VIG system is Pareto Race, a dynamic and visual approach for exploring the efficient frontier of a multiple objective linear programming problem. Our objective is to extend Pareto Race to large-scale multiple objective linear programming. The approach works with any efficient solutions that are in general not extreme point solutions. Interactive use of computer graphics plays a central role. The approach, the underlying theory, and an illustrative example are described.     

On external estimates for reachable sets of nonlinear control systems

The paper is devoted to the problem of constructing external estimates for reachable sets of a nonlinear control system. The estimates are constructed in the form of level sets of smooth functions in the space of states satisfying differential inequalities. In the system under consideration, the linear part is found, for which the corresponding functions are assumed to be known. The method proposed for estimating trajectories of a nonlinear system is based on modifying estimates for the linear part and on applying the comparison principle.     

Nonlinear stabilization of abstract systems via a linear observability inequality and application to vibrating PDE's

This Note is concerned with the links between nonlinear stabilization of hyperbolic systems and linear observability for the unforced corresponding linear system, for locally distributed and boundary feedbacks as well. We show that if the linear system is observable through a locally distributed (resp. boundary) observation, then any dissipative nonlinear feedback locally distributed (resp. active only on a part of the boundary) stabilize the system and we give a general energy decay formula. Our results generalize previous results by Haraux (1989) and Ammari and Tucsnak (2001) for linear feedbacks. We show by this way that for the locally distributed case, one can combine the optimal geometric conditions of Bardos et al. (1992) and the method of Alabau-Boussouira (2005) to deduce energy decay rates for nonlinear damped systems.     

Scalar labelings for homotopy paths

Two scalar labelings are introduced for obtaining approximate solutions to systems of nonlinear equations by simplicial approximation. Under reasonable assumptions the new scalar-labeling algorithms are shown to follow, in a limiting sense, homotopy paths which can also be tracked by piecewise linear vector labeling algorithms. Though the new algorithms eliminate the need to pivot on a system of linear equations, the question of relative computational efficiency is unresolved.The work of this author was supported in part by NSF Grant No. MCS-77-15509 and by ARO Grant No. DAAG-29-78-G-0160.The work of this author was supported in part by ONR Grant No. N00014-75-C-0495 and NSF Grant No. 81058.     

Asymptotical time response of time varying state space dynamic using homotopy perturbation method

The main goal of this study is to present a technique to find a state response of multivariable time varying systems. In this paper a novel Homotopy Perturbation based Method (HPM) will be presented to find a dynamic response of time varying system. According to this method, the linear part of the described system is partitioned into two time varying and invariant subsections. Time invariant part analytically constructs the state transition matrix. This matrix is a core of the rest of time varying differential equation without any further changes in a sequence order. The main advantage of this method is only the necessity to solve the time invariant part of the state transition matrix. Simulation results verify the significance of the proposed analytic and asymptotic method.     

Frequency characteristics of flexibly vibration discrete-continuous mechatronic system

The purpose of this paper is formulating of problem of flexibly vibrating mechatronic system. The main approach of the subject was to formulate the problem in the form of set of differential equation of motion and state equation of considered mechatronic model of object. The considered flexibly vibrating mechanical system is a continuous beam, clamped at one of its end. Integral part of mechatronic system is a transducer, extorted by harmonic voltage. In the paper the linear mechanical subsystem and linear electric subsystem of mechatronic system has been considered. The methods of analysis and obtained results can be base on design and investigation for this type of mechatronic systems. The mechatronic system formed from mechanical and electric subsystems with electromechanical bondage has been considered. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A note on analyticity to piezoelectric systems

In this paper, we consider a piezoelectric dissipative system in which a dissipation term is given through the mechanical part of the stress tensor. The abstract Cauchy problem associated to this system generates a semigroup of linear operators. We establish the analyticity of this semigroup that, in turn, implies the exponential decay of the corresponding energy and the strong regularity of the solutions. Copyright © 2012 John Wiley & Sons, Ltd.     

关于三阶变系数线性微分方程解的不稳定性

文献[1]用李雅普诺夫第二方法证明了特征根均具有负实部的缓变系数动力系统的渐近稳定性.本文也用李雅普诺夫第二方法给出至少有一个特征根具有正实部的三阶变系数线性微分方程解的不稳定性的充分条件.