The relation between problems of pursuit,controllability and stability in the large in linear systems with different types of constraints

A relation is established between problems of pursuit, controllability and stability in the large in linear systems when a geometric constraint is imposed on the control vector of the pursuer and an integral constraint is imposed on the control function of the evader.     

Robust controllability of linear systems: Analysis and application to pursuit problems

Robust controllability conditions for a linear uncertain system are derived. A class of transferring strategies is obtained. A robust controllability set, respecting a given control constraint, is constructed. An application example is presented. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

On different types of stability of linear polytopic systems

Different types of stability of linear uncertain systems are considered. Stability conditions in terms of nonsingularity are obtained. Numerical examples of using the Bernstein expansion of the determinant function are given.     

Complete controllability of perturbed linear control systems,revisited

LetL ¹(I, ^n,n )×M(I, ^n,m ) be the space of all pairs (A, B), whereA andB are measurable functions from a compact intervalI to ^n,n and ^n,m , respectively, andA is Lebesgue integrable. Also, let this space be endowed with the topology of theL ¹-norm with respect toA and the topology of convergence in measure with respect toB. Then, the set of all pairs (A, B), for which the corresponding linear control system

Numerical solution of linear least-squares problems with linear equality constraints

In Ref. 1, a perturbation theory for the linear least-squares problem with linear equality constraints is presented. In this paper, the condition numbers of a general formula given in Ref. 1 are examined in order to compare them with the condition numbers of the two matrices of the problem. A class of test problems is also defined to study experimentally the numerical stability of three algorithms.This work was partially supported by the Ministero della Pubblica Istruzione, Rome, Italy.The authors thank the Centro Interdipartimentale di Calcolo Automatico e di Informatica Applicata of the University of Modena for having provided computing time on its VAX computer.     

Stability by linear approximation and the relation between the stability of difference and differential fractional systems

The paper is devoted to study the stability of nonlinear fractional order difference systems by their linear approximation. Additionally, we show the relation between the stability of linear fractional order differential systems and their discretizations. Copyright © 2016 John Wiley & Sons, Ltd.     

Finite controllability of linear parabolic systems with stationary final states

The aim of controllability is usually to reach a final state only for a moment. But for practical reasons the trajectory often has to stay in this position for some time. The set of stationary states is analized and controllability for stationary states is investigated with finite dimensional controls.     

On constraint controllability of linear systems in Banach spaces

In this paper, we give an addendum to a result of Dolecki and Russell (Ref. 1) related to the duality relationship between observation and control for linear systems in Banach spaces. Our results relate the controllability of a system to the constraint controllability of that system and to the observability of an adjoint system. The main tool used here is an extension of the classical open mapping theorem.     

A paradox in linear fractional transportation problems with mixed constraints

The present paper studies a paradox in Linear Fractional Transportation Problems with 'mixed constraints'. A sufficient condition for the existence of a paradox is established. Paradoxical range of flow is obtained for any flow in which the corresponding objective function value is less than that of the original Linear Fractional Transportation Problem with 'mixed constraints'     

The method of pursuit by several controlled objects of different types

The problem of the pursuit of one evader by several controlled objects of different types is examined. The sufficient conditions are obtained for the pursuit game to terminate in a finite time. The proposed method of pursuer interaction assumes that the pursuing players are separated into two groups, the first of which holds the evader in some domain, while the second searches for the evader in this domain. The paper touches on the researches in /1–9/. Typical examples illustrate the results.     

On the optimal control of linear systems with linear performance criterion and constraints

We suggest an analytical-numerical method for solving a boundary value optimal control problem with state, integral, and control constraints. The embedding principle underlying the method is based on the general solution of a Fredholm integral equation of the first kind and its analytic representation; the method permits one to reduce the boundary value optimal control problem with constraints to an optimization problem with free right end of the trajectory.     

The stability of linear potential gyroscopic systems

The stability of linear potential systems with a degenerate matrix of gyroscopic forces is investigated. Particular attention is devoted to the case of three degrees of freedom. In a development of existing results [Kozlov VV. Gyroscopic stabilization and parametric resonance. Prikl. Mat. Mekh. 2001; 65(5): 739–745], the sufficient conditions for gyroscopic stability are obtained. An algorithm for applying these conditions is proposed using the example of the problem of the motion of two mutually gravitating bodies, each of them being modelled by two equal point masses, connected by weightless inextensible rods.     

On weak and strong reachability and controllability of infinite-dimensional linear systems

The notions of reachability and controllability generalize to infinite-dimensional systems in two different ways. We show that the strong notions are equivalent to finite-time reachability and controllability. For discrete systems in Hilbert space, we get simple relations generalizing the Kalman conditions. In the case of a continuous system in Hilbert space, weak reachability is equivalent to the weak reachability of a related discrete system via the Cayley transform.This research was partially supported by the Batsheva de Rothschild Fund for the Advancement of Science and Technology.     

随机时滞混杂区间系统的鲁棒稳定性

利用Lyapunov泛函方法及线性矩阵不等式，得到了随机时滞混杂区间系统的能失性及鲁棒稳定性的充分判据。     

Completely positive reformulations of polynomial optimization problems with linear constraints

A polynomial optimization problem (POP) is an optimization problem in which both the objective and constraints can be written in terms of polynomials on the decision variables. Recently, it has been shown that under appropriate assumptions POPs can be reformulated as conic problems over the cone of completely positive tensors; which generalize the set of completely positive matrices. Here, we show that by explicitly handling the linear constraints in the formulation of the POP, one obtains a generalization of the completely positive reformulation of quadratically constrained quadratic programs recently introduced by Bai et al. (Math Program 1–28, 2016).     

Module structure of constant linear systems and its applications to controllability

We shall introduce a new module structure to a large class of continuous-time constant linear systems. This is done as a natural extension of the classical k[z]-module structure of finite-dimensional constant linear systems. This module action is used to investigate the relationship between reachability and controllability of linear systems. After introducing the notion of K-controllability due to Kamen [12], we give the following result in Section 5: If a constant linear system is described by a functional differential equation $\text{x}\text{?}\text{= Fx + Gu,}\text{where}\text{x}\text{and}\text{G}$ belong to a Banach space X, and if G is K-controllable to zero, then every reachable state is reachable and controllable in bounded time. (The result given in Section 5 is a little more general than this.) We also give a simple example in Section 6 to illustrate this result.