The planar discontinuous piecewise linear refracting systems have at most one limit cycle

In this paper we investigate the limit cycles of planar piecewise linear differential systems with two zones separated by a straight line. It is well known that when these systems are continuous they can exhibit at most one limit cycle, while when they are discontinuous the question about maximum number of limit cycles that they can exhibit is still open. For these last systems there are examples exhibiting three limit cycles.The aim of this paper is to study the number of limit cycles for a special kind of planar discontinuous piecewise linear differential systems with two zones separated by a straight line which are known as refracting systems. First we obtain the existence and uniqueness of limit cycles for refracting systems of focus-node type. Second we prove that refracting systems of focus–focus type have at most one limit cycle, thus we give a positive answer to a conjecture on the uniqueness of limit cycle stated by Freire, Ponce and Torres in Freire et al. (2013). These two results complete the proof that any refracting system has at most one limit cycle.     

Two Limit Cycles in Liénard Piecewise Linear Differential Systems

Some techniques for studying the existence of limit cycles for smooth differential systems are extended to continuous piecewise linear differential systems. Rigorous new results are provided on the existence of two limit cycles surrounding the equilibrium point at the origin for systems with three zones separated by two parallel straight lines without symmetry. As a relevant application, it is shown the existence of bistable regimes in an asymmetric memristor-based electronic oscillator.

     

Hopf bifurcation at infinity in 3D symmetric piecewise linear systems. Application to a Bonhoeffer–van der Pol oscillator

In this work, a Hopf bifurcation at infinity in three-dimensional symmetric continuous piecewise linear systems with three zones is analyzed. By adapting the so-called closing equations method, which constitutes a suitable technique to detect limit cycles bifurcation in piecewise linear systems, we give for the first time a complete characterization of the existence and stability of the limit cycle of large amplitude that bifurcates from the point at infinity. Analytical expressions for the period and amplitude of the bifurcating limit cycles are obtained. As an application of these results, we study the appearance of a large amplitude limit cycle in a Bonhoeffer–van der Pol oscillator.     

Limit cycles of piecewise linear dynamical systems with three zones and lateral systems

In this paper, we give some evidences what cause more limit cycles for piecewise dynamical systems. We say, the angles or the number of zones are critical points. We study an example of linear lateral systems and an example of linear Y-shape systems, and prove that they have five and four crossing limit cycles by using Newton-Kantorovich Theorem, respectively.     

Limit cycle bifurcations of planar piecewise differential systems with three zones

This paper considers the limit cycle bifurcation problem of planar piecewise differential systems with three zones. Some computation formulas studied the problem of limit cycle bifurcations are provided by introducing multiple parameters. As an application to the obtained method, the number of limit cycles of a piecewise linear system with three zones studied in Lima et al. (2017) is discussed and some more limit cycles are found.     

Phase portraits of piecewise linear continuous differential systems with two zones separated by a straight line

This paper provides the classification of the phase portraits in the Poincaré disc of all piecewise linear continuous differential systems with two zones separated by a straight line having a unique finite singular point which is a node or a focus. The sufficient and necessary conditions for existence and uniqueness of limit cycles are also given.     

Controllability and observability of coordinated linear systems

This paper develops concepts of controllability and observability for the class of coordinated linear systems, a special class of linear hierarchical systems with a top-to-bottom information structure. The usual concepts of reachability and indistinguishability for linear systems are refined, taking into account the decentralized nature of coordinated linear systems. The corresponding controllability and observability decompositions are derived, and several concepts of controllability and observability for coordinated linear systems are introduced and discussed. These results are then combined to a definition and characterization of stabilizability via dynamic measurement feedback for this class of systems.     

Numerical solution of nonlinear inverse coefficient problems for ordinary differential equations

Parametric identification for a class of nonlinear objects with lumped parameters described by systems of ordinary differential equations is studied. The problem is to recover the coefficients of a dynamical system depending on the phase state. For that purpose, the phase space is subdivided into a finite set of subsets or zones in which the coefficients are assumed to be constant or linear functions of state. Once the coefficients in such a form are obtained, interpolation and approximation can be used to represent the coefficients as functions of the phase variables.     

A Note on the Bendixson-Dulac Theorem for Refracted Systems with Multiple Zones

This note extends the Bendixson-Dulac theorem to refracted systems with multiple zones. As an application, we prove that piecewise linear Duffing-type system has neither crossing limit cycles nor sliding limit cycles Therefore, it gives a positive answer to the Conjecture of \cite{TA2007}.     

Modelling structured geothermal systems II: Application of linear stability analysis

This paper illustrates how information obtained from linearised mathematical models can be used to investigate the development in time of complex natural systems such as structured geothermal reservoirs. Solutions to the saturation equation resulting from a linear stability analysis are obtained in the form of unidirectional waves. From the direction of propagation of these waves, the upper layers of a vapour-dominated zone equilibrate earlier than the lower layers, and the lower layers of a liquid-dominated zone equilibrate earlier than the upper layers. Our results support the two-layer models suggested for Wairakei (New Zealand), The Geysers (California), and Larderello (Italy). They also explain why the inverse model with a liquid layer over a vapour-dominated zone proposed for the Kawah Kamojang system (Indonesia) is not valid. We suggest to replace this model by a single-phase liquid layer over a liquid-dominated convective zone. The latter two-layer structure as well as the three-layer structure with a liquid layer between the upper vapour and lower liquid-dominated zones are proven to be possible in a homogeneous reservoir. Computer simulation results are presented.     

A QMR-based interior-point algorithm for solving linear programs

A new approach for the implementation of interior-point methods for solving linear programs is proposed. Its main feature is the iterative solution of the symmetric, but highly indefinite 2×2-block systems of linear equations that arise within the interior-point algorithm. These linear systems are solved by a symmetric variant of the quasi-minimal residual (QMR) algorithm, which is an iterative solver for general linear systems. The symmetric QMR algorithm can be combined with indefinite preconditioners, which is crucial for the efficient solution of highly indefinite linear systems, yet it still fully exploits the symmetry of the linear systems to be solved. To support the use of the symmetric QMR iteration, a novel stable reduction of the original unsymmetric 3×3-block systems to symmetric 2×2-block systems is introduced, and a measure for a low relative accuracy for the solution of these linear systems within the interior-point algorithm is proposed. Some indefinite preconditioners are discussed. Finally, we report results of a few preliminary numerical experiments to illustrate the features of the new approach.     

The qualitative analysis of a class of planar Filippov systems

We use the theory of differential inclusions, Filippov transformations and some appropriate Poincaré maps to discuss the special case of two-dimensional discontinuous piecewise linear differential systems with two zones. This analysis applies to uniqueness and non-uniqueness for the initial value problem, stability of stationary points, sliding motion solutions, number of closed trajectories, existence of heteroclinic trajectories connecting two saddle points forming a heteroclinic cycle and existence of the homoclinic trajectory     

Stability of linear time-varying delay systems and applications to control problems

This paper deals with the stability of a class of linear time-varying systems with multiple delays. Using the Lyapunov function method, we give sufficient delay-dependent conditions for the exponential stability with a given convergence rate, which are described in terms of linear matrix inequalities (LMI) and the solution of Riccati differential equations (RDE). The results are applied to the problem of stabilization of linear time-varying control systems with multiple delays. Numerical examples are given to illustrate the results.     

Dwell-time stability and stabilization conditions for linear positive impulsive and switched systems

Several results regarding the stability and the stabilization of linear impulsive positive systems under arbitrary, constant, minimum, maximum and range dwell-time are obtained. The proposed stability conditions characterize the pointwise decrease of a linear copositive Lyapunov function and are formulated in terms of finite-dimensional or semi-infinite linear programs. To be applicable to uncertain systems and to control design, a lifting approach introducing a clock-variable is then considered in order to make the conditions affine in the matrices of the system. The resulting stability and stabilization conditions are stated as infinite-dimensional linear programs for which three asymptotically exact computational methods are proposed and compared with each other on numerical examples. Similar results are then obtained for linear positive switched systems by exploiting the possibility of reformulating a switched system as an impulsive system. Some existing stability conditions are retrieved and extended to stabilization using the proposed lifting approach. Several examples are finally given for illustration.     

离散系统Lyapunov第二方法的一些问题

本文讨论线性时不变离散系统Lyapunov方程解集的几何性质以及分段线性离散系统的稳定性,得出每个子系统都是稳定的分段线性离散系统渐近稳定的一些充分条件,并把这些结果应用于二阶分段线性系统.     

Stochastic stability of positive Markov jump linear systems with fixed dwell time

This paper studies the stochastic stability of positive Markov jump linear systems with a fixed dwell time. By constructing an auxiliary system that originated from the initial system with state jumps, sufficient and necessary conditions of stochastic stability for positive Markov jump linear systems are obtained with both exactly known and partially known transition rates. The main idea in the latter case is applying a convex combination to convert bilinear programming into linear programming problems. On this basis, multiple piecewise linear co-positive Lyapunov functions are provided to achieve less conservative results. Then state feedback controller is designed to stabilize the positive Markov jump linear systems by solving linear programming problems. Numerical examples are presented to illustrate the viability of our conclusions.     

Transformations of interval linear systems of equations and inequalities

Very recently, new results on transformation of interval linear systems and on generalizations of the Farkas lemma to interval systems appeared in the literature. They are by far not obvious since the standard transformations on linear systems are not easily adapted to interval system due to the dependency problem. The aim of this paper was to come up with new possible transformations and to extend the results to more general classes of AE interval systems and to linear parametric systems. We also show that the transformations can help in simplifying the proofs of some characterization theorems.     

Analysis of stability and synthesis of stable control systems for a class of nonlinear nonstationary systems

On the basis of sufficient condition of the stability of linear nonstationary discrete systems stated in this paper, we carry out an analysis of stability for a special class of discrete systems with nonstationary linear part, whose parameters satisfy constraints in the form of sets, and with a scalar nonlinear function satisfying linear or nonlinear restrictions. The problem of parametric synthesis of robustly stable control systems is solved for the same class of objects. The obtained results are generalized to the class of nonstationary systems with many nonlinearities of the same type.     

无重叠区的双抓钩周期性排序问题的求解

在当今的自动化制造系统中，计算机控制的抓钩的排序直接影响系统的生产率。本文研究了产品在系统的一边装载、而在另一边卸载的电镀线周期性排序问题。工件在每个工作站的处理时间在给定时间范围内，工作站之间没有缓冲槽，相同轨道上的两个抓钩用于工作站之间工件的运送，目标是对运送进行排序以极小化生产周期。为了求解这个问题，本文提出一个求解方法，所提出的方法首先将生产线分为两个无重叠的区域，并且为每个区域分配一个抓钩，然后，提出了一个给定抓钩分配下的混合整数线性规划模型。通过求解不同抓钩分配下模型的最优解，并且选择这些解中最好的一个，以便得到最优解，一个标杆示例被运行，以表明该方法的应用。另外，给出有多重处理槽工序问题的模型和求解方法。     

Solving Symmetric Arrowhead and Special Tridiagonal Linear Systems by Fast Approximate Inverse Preconditioning

A new class of approximate inverses for arrowhead and special tridiagonal linear systems, based on the concept of sparse approximate Choleski-type factorization procedures, are introduced for computing fast explicit approximate inverses. Explicit preconditioned iterative schemes in conjunction with approximate inverse matrix techniques are presented for the efficient solution of symmetric linear systems. A theorem on the rate of convergence of the explicit preconditioned conjugate gradient scheme is given and estimates of the computational complexity are presented. Applications of the proposed method on linear and nonlinear systems are discussed and numerical results are given.