Linear Complementarity Systems Research Articles

Bifurcation analysis of a non-smooth prey–predator model by a differential linear complementarity system

The aim of this paper is to investigate a prey–predator system with threshold harvesting for both species, whereas most researchers are devoted to studying a threshold policy for one population to avoid intractable mathematical simulation occurred by their coupling. Due to the equivalence between differential inclusion and differential complementarity system, we reformulate this ecosystem as a differential linear complementarity system. Then specialized algorithms for the complementarity problem allow us to handle such non-smooth structure, thereby performing a numerical examination of the dynamics and bifurcations of our proposed system with success. As a result, we can observe that this system exhibits many peculiar bifurcation patterns that are inherent to a non-smooth dynamic system, including boundary node bifurcation, pseudo-saddle–node bifurcation, touching bifurcation, and sliding homoclinic bifurcation. Specifically, we observe a multiple crossing bifurcation that results from the superposition of a sliding homoclinic bifurcation and a boundary saddle bifurcation. They are originated from the coupling of their own independent non-smooth structures of the two species. This system also admits some conventional bifurcations like saddle–node bifurcation, Hopf bifurcation and Bogdanov–Takens bifurcation, which are identified previously in a prey–predator system without threshold policy. Both the theoretical and numerical results indicate that the non-smooth structure of the threshold harvesting policy increases equilibria, as well as the dynamical complications of the system.

Exponential Time-Stepping Method for Linear Complementarity Systems

Linear complementarity system (LCS) consists of an ordinary differential equation (ODE) and a linear complementarity problem (LCP). In this article, we propose an exponential time-stepping method for solving LCS, which uses exponential integrator to discretize the ODE and solves the LCPs at the discrete time points. Numerical results are reported for illustrating its good performance.

Analysis of the Implicit Euler Time-Discretization of Semiexplicit Differential-Algebraic Linear Complementarity Systems

This article is largely concerned with the time-discretization of differential-algebraic equations (DAEs) with complementarity constraints, which we name differential-algebraic linear complementarity systems (DALCSs). Specifically, the Euler implicit discretization of DALCSs is analyzed: the one-step nonsmooth problem, which is a generalized equation, is shown to be well-posed under some conditions; then the convergence of the discretized solutions is studied, and the existence of solutions to the continuous-time system is shown as a consequence. Passivity of some operators is pivotal to the analysis. Examples from circuits, mechanics, and switching DAEs illustrate the applicability of the developments.

Analysis of the implicit Euler time-discretization of passive linear descriptor complementarity systems

paper 7269 This article is largely concerned with the time-discretization of descriptor-variable systems coupled to with complementarity constraints. They are named descriptor-variable linear complementarity systems (DVLCS). More speci cally passive DVLCS with minimal state space representation are studied. The Euler implicit discretization of DVLCS is analysed: the one-step non-smooth problem (OSNSP), that is a generalized equation, is shown to be well-posed under some conditions. Then the convergence of the discretized solutions is studied. Several examples illustrate the applicability and the limitations of the developments.

State observers for the time discretization of a class of impulsive mechanical systems

AbstractIn this work, we investigate the state observer problem for linear mechanical systems with a single unilateral constraint, for which neither the impact time instants nor the contact distance is explicitly measured. We propose to attack the observer problem by transforming and approximating the original continuous‐time system by a discrete linear complementarity system (LCS) through the use of the Schatzman–Paoli scheme. From there, we derive a deadbeat observer in the form of a linear complementarity problem. Sufficient conditions guaranteeing the uniqueness of its solution then serve as observability conditions. In addition, the discrete adaptation of an existing passivity‐based observer design for LCSs can be applied. A key point in using a time discretization is that the discretization acts as a regularization, that is, the impacts take place over multiple time steps (here two time steps). This makes it possible to render the estimation error dynamics asymptotically stable. Furthermore, the so‐called peaking phenomenon appears as singularity within the time discretization approach, posing a challenge for robust observer design.

Analyzing Feedback Interconnections of Maximal Monotone Systems Using Dissipativity Approach

We consider interconnections of two dynamical systems in feedback configuration. The dynamics of the individual systems are modeled by a differential inclusion, and the corresponding set-valued mapping is (anti-) maximal monotone with respect to the state of the system for each fixed value of the external signal that defines the interconnection. We provide conditions on these mappings under which the dynamics of the resulting interconnected system are (anti-) maximal monotone. An interpretation of our main result is provided: firstly, by considering dynamical systems defined by the gradient of a saddle function, and secondly, by considering an interconnection of incrementally passive systems. In the same spirit, when we associate more structure to the individual systems by considering linear complementarity systems, we allow for more flexibility in describing the interconnections and derive more specific sufficient conditions in terms of system matrices that result in the overall system being described by (anti-) maximal monotone operator.

Simulation-Based Study of Biological Systems with Threshold Policy by a Differential Linear Complementarity System

Threshold policy is more realistic than continuous control for biological system management. Most related works are devoted to studying a single-threshold value for one single population, thereby avoiding complicated mathematical analysis of the nonsmooth differential equations. Based on the fact that numerical simulations play an important role in analyzing and understanding the intrinsic mechanism of a biological experiment and system, we hereby propose a differential linear complementarity system to reformulate the biological system with threshold policy. Using this method, we can transform a biological system with multiple-threshold values for one or more population to a differential linear complementarity system, where the corresponding dynamics can be investigated numerically by various algorithms for the complementarity problem. Firstly, the well-posedness of solutions of the differential linear complementarity system and its discretized method are derived explicitly. Then we illustrate the application of our approach to two systems which are a population harvesting system with threshold policy and an HIV replication system with threshold therapy, respectively. Numerical results demonstrate that those nonsmooth biological systems exhibit much more complex dynamics than the corresponding smooth systems. These results also validate the effectiveness and simplicity of the method that reformulates a common biological system with multiple-threshold policy by a differential linear complementarity system.

Uniqueness and Hyers-Ulam Stability of Random Differential Variational Inequalities with Nonlocal Boundary Conditions

In this paper, we consider a new class of random differential variational inequalities (RDVIs) with nonlocal boundary conditions in Hilbert spaces. We apply the projection operator, Gronwall’s lemma and a result on the existence of a random differential inclusion to establish uniqueness and Hyers–Ulam stability results of the abstract inequality. As an illustrative application, the linear random differential complementarity systems and a price control model are investigated.

A New MIP Approach on the Least Distance Problem in DEA

In this paper, we deal with the least distance problem (LDP) in Data Envelopment Analysis (DEA), which is to find a closest efficient target over the whole efficient frontier. To this end, we define the efficient frontier by a linear complementarity system and propose a mixed integer programming (MIP) approach to solve the LDP. Our proposed MIP approach: (1) can solve the LDP based on [Formula: see text]-norm ([Formula: see text]) by using a state-of-the-art solver and obtain the closest efficient target over the whole efficient frontier instead of a subset of it; (2) can be applied for computing the least distance DEA models satisfying the monotonicity; (3) is more user-friendly, because it allows a decision maker to improve the efficiency of a decision making unit (DMU) by setting the affordable input/output level under his/her circumstance. Thus, the efficient target provided by our approach may be more appropriate from the perspective of the decision makers of DMUs.

Convergence of discrete approximation for differential linear stochastic complementarity systems

In this paper, we investigate a class of differential linear stochastic complementarity system consisting of an ordinary differential equation and a stochastic complementarity problem. The existence of solutions for such system is obtained under two cases of the coefficient matrix of the linear stochastic complementarity problem: P-matrix and positive semi-definite matrix. As for the first case, the sample average approximate method and time-stepping method are adopted to get the numerical solutions. Furthermore, a regularization approximation is introduced to the second case to ensure the uniqueness of solutions. The corresponding convergence analysis is conducted, and numerical examples are presented to illustrate the convergence results we derived. Finally, we provide numerical results which come from applications involving dynamic traffic flow problems to support our theorems.

A generalized Newton method for a class of discrete-time linear complementarity systems

In this paper, we propose a generalized Newton method for solving a class of discrete-time linear complementarity systems consisting of a system of linear equations and a linear complementarity constraints with a Z-matrix. We obtain a complete characterization of the least element solution of a linear complementarity problem with a Z-matrix that a solution is the least element solution if and only if the principal submatrix corresponding to the nonzero components of the solution is an M-matrix. We present a Newton method for solving a linear complementarity problem with a Z-matrix. We propose a generalized Newton method for solving the discrete-time linear complementarity system where the linear complementarity problem constraint is solved by the proposed Newton method. Under suitable conditions, we show that the generalized Newton method converges globally and finds a solution in finitely many iterations. Preliminary numerical results show the efficiency of the proposed method.

Linear Complementarity Systems and Cone-Copositive Lyapunov Stability

Exponential stability of the origin of linear complementarity systems (LCS) is analyzed by applying Lyapunov theory. By representing the feasibility and the solution sets of the LCS as cones, a cone-copositive approach is used to get sufficient stability conditions expressed in terms of linear matrix inequalities (LMI). The proposed method is constructive in the sense that the solution of the set of LMI directly provides a quadratic Lyapunov function. Sufficient conditions for piecewise quadratic Lyapunov functions are obtained, as well. Illustrative examples show the effectiveness of the approach.

On the Controllability and Stabilizability of Linear Complementarity Systems

Abstract This paper studies controllability and stabilizability of linear complementarity systems that can be cast as continuous piecewise affine dynamical systems. Under a certain right-invertibility assumption, we present a la Hautus necessary and sufficient conditions for both controllability and stabilizability.

Least-Element Time-Stepping Methods for Simulation of Linear Networks with Ideal Switches

Linear networks with ideal switches have various applications in power converters, signal processing and control problems, which can be modeled by linear complementarity systems (LCSs). This paper presents new results on the least-element time-stepping method for simulation of linear networks with ideal switches for a class of LCSs. The method is efficient and stable and can be easily implemented. The convergence results and preliminary numerical results show that the least-element time-stepping method is efficient for verifying accuracy of approximate solutions.

Uniqueness and Hyers–Ulam Stability Results for Differential Variational Inequalities with Nonlocal Conditions

In this paper, we study the nonlocal problem for differential variational inequalities (DVIs) in Hilbert spaces. We proved that under some suitable conditions the considered problem is Hyers–Ulam stable and has a unique solution. It is shown how the abstract results can be applied to study the linear differential complementarity systems and a price control model. Some examples are given to illustrate the obtained results.

Quadratic Optimal Control of Linear Complementarity Systems: First-Order Necessary Conditions and Numerical Analysis

This article is dedicated to the analysis of quadratic optimal control of linear complementarity systems (LCSs), which are a class of strongly nonlinear and nonsmooth dynamical systems. Necessary first-order conditions are derived that take the form of an LCS with inequality constraints, hence are numerically tractable. Sufficiency is proved in a specific sense. Numerical examples illustrate the theoretical developments and demonstrate the efficiency of the complementarity approach.

An exponential integrator-based discontinuous Galerkin method for linear complementarity systems

An exponential integrator-based discontinuous Galerkin method for linear complementarity systems

Extragradient methods for differential variational inequality problems and linear complementarity systems

In this paper, 2 extragradient methods for solving differential variational inequality (DVI) problems are presented, and the convergence conditions are derived. It is shown that the presented extragradient methods have weaker convergence conditions in comparison with the basic fixed‐point algorithm for solving DVIs. Then the linear complementarity systems, as an important and practical special case of DVIs, are considered, and the convergence conditions of the presented extragradient methods are adapted for them. In addition, an upper bound for the Lipschitz constant of linear complementarity systems is introduced. This upper bound can be used for adjusting the parameters of the extragradient methods, to accelerate the convergence speed. Finally, 4 illustrative examples are considered to support the theoretical results.

Stability Analysis of Equilibria of Linear Delay Complementarity Systems

We present stability criteria for equilibria of a class of linear complementarity systems, subjected to discrete and distributed delay. We present necessary and sufficient conditions for local exponential stability, inferred from the spectrum location of a corresponding system of delay differential algebraic equations. Subsequently, we obtain sufficient LMI-based conditions for global asymptotic stability using Lyapunov–Krasovskii functionals.

Prelimenary results on the optimal control of linear complementarity systems

In this paper, we give some results concerning the quadratic optimal control of linear complementarity systems. We derive first order conditions for this system, that we manage to express as a Mixed Linear Complementarity System. We then use this result to build numerical schemes, which are expressed as Mathematical Problems with Equilibrium Constraints.