Analysis of switched linear systems in the plane,part 2: Global behavior of trajectories,controllability and attainability

The controllability and attainability properties of switched linear systems in the plane are investigated. A main result is the state-space decomposition theorem which classifies various convex regions inR ² according to their controllability properties. A preliminary investigation into the problem of determining minimum switch trajectories between two points inR ² is also presented.This research was supported by the Office of Naval Research, ONR Contract No. N0014-80-C-0199, and the United States Department of Energy, DOE Contract No. DE-AC01-79-ET-29363.     

Analysis of switched linear systems in the plane,part 1: Local behavior of trajectories and local cycle geometry

This paper investigates the problem of control of switched linear systems evolving inR ². The concept of an opposition point is introduced, and its properties related to the existence of closed trajectories in the phase plane are investigated. The geometry of cycles in a neighborhood of an opposition point is also studied.This research was supported by the Office of Naval Research, ONR Contract No. N0013-80-C-0199, and the United States Department of Energy, DOE Contract No. DE-AC01-79-ET-29363.     

Sharp energy estimates for nonlinearly locally damped PDEs via observability for the associated undamped system

We consider the problem of sharp energy decay rates for nonlinearly damped abstract infinite-dimensional systems. Direct methods for nonlinear stabilization generally rely on multiplier techniques, and thus are valid under restrictive geometric conditions compared to the optimal geometric optics condition of Bardos et al. (1992) [10]. We prove sharp, simple and quasi-optimal energy decay rates through an indirect method, namely an observability estimate for the corresponding undamped system. One of the main advantage of these results is that they allow to combine optimal geometric conditions, as for instance that of Bardos et al. (1992) [10] and the optimal-weight convexity method of the first author (Alabau-Boussouira, 2010 [6], Alabau-Boussouira, 2005 [2]) to deduce very simple and quasi-optimal energy decay rates for nonlinearly locally damped systems. We also show that using arguments based on Russell's principle (Russell, 1978 [24]), one can deduce sharp energy decay rates from the exponential stabilization of the linearly damped system. Our results extend to nonlinearly damped systems, those of Haraux (1989) [14] and Ammari and Tucsnak (2001) [9] which concern linearly damped systems.     

Change-of-bases abstractions for non-linear hybrid systems

We present abstraction techniques that transform a given non-linear dynamical system into a linear system, or more generally, an algebraic system described by polynomials of bounded degree, so that invariant properties of the resulting abstraction can be used to infer invariants for the original system. The abstraction techniques rely on a change-of-bases transformation that associates each state variable of the abstract system with a function involving the state variables of the original system. We present conditions under which a given change-of-bases transformation for a non-linear system can define an abstraction. Furthermore, the techniques developed here apply to continuous systems defined by Ordinary Differential Equations (ODEs), discrete systems defined by transition systems and hybrid systems that combine continuous as well as discrete subsystems.The techniques presented here allow us to discover, given a non-linear system, if a change-of-bases transformation involving degree-bounded polynomials yielding an algebraic abstraction exists. If so, our technique yields the resulting abstract system, as well. Our techniques enable the use of analysis techniques for linear systems to infer invariants for non-linear systems. We present preliminary evidence of the practical feasibility of our ideas using a prototype implementation.     

Comparison of path-complete Lyapunov functions via template-dependent lifts

This paper investigates, in the context of discrete-time switched systems, the problem of comparison for path-complete stability certificates. We introduce and study abstract operations on path-complete graphs, called lifts, which allow us to recover previous results in a general framework. Moreover, this approach highlights the existing relations between the analytical properties of the chosen set of candidate Lyapunov functions (the template) and the admissibility of certain lifts. This provides a new methodology for the characterization of the ordering relation of path-complete Lyapunov functions criteria, when a particular template is chosen. We apply our results to specific templates, notably the sets of primal and dual copositive norms, providing new stability certificates for positive switched systems. These tools are finally illustrated with the aim of numerical examples.     

Bilinear Control and Application to Flexible a.c. Transmission Systems

The purpose of this paper is an integrated overview of bilinear systems (BLS) research which has evolved over the past few decades, and a new result on control of flexible a.c. transmission systems (FACTS) is presented. BLS may be derived in many cases from principles of physics, chemistry, biology, socioeconomics, and engineering. In other cases, BLS are more accurate approximations to nonlinear systems than are traditional linear systems, as shown for example by the added bilinear terms (in state and control) for the Taylor series.While an appropriately designed linear control system may be optimum relative to some quadratic performance index without added constraints, bilinear or parametric control can be designed to improve more global performance and indeed to increase the region of attainable states. Such controllability and stabilization of BLS and of a series line-capacitor controlled FACTS is presented.     

Preconditioning discretizations of systems of partial differential equations

This survey paper is based on three talks given by the second author at the London Mathematical Society Durham Symposium on Computational Linear Algebra for Partial Differential Equations in the summer of 2008. The main focus will be on an abstract approach to the construction of preconditioners for symmetric linear systems in a Hilbert space setting. Typical examples that are covered by this theory are systems of partial differential equations which correspond to saddle point problems. We will argue that the mapping properties of the coefficient operators suggest that block diagonal preconditioners are natural choices for these systems. To illustrate our approach a number of examples will be considered. In particular, parameter‐dependent systems arising in areas like incompressible flow, linear elasticity, and optimal control theory will be studied. The paper contains analysis of several models which have previously been discussed in the literature. However, here each example is discussed with reference to a more unified abstract approach. Copyright © 2010 John Wiley & Sons, Ltd.     

Supervisory control of hybrid systems based on model abstraction and guided search

This paper addresses the task of computing supervisory controllers by which hybrid systems with nonlinear continuous dynamics are driven into goal sets while safety specifications are met. For this class of systems, the (conservatively approximative) determination of reachable states is an important but also a computationally expensive step of the controller synthesis. This contribution proposes a technique aiming at reducing the reach set computation by using abstract models and guided search. For a discrete abstraction of the hybrid model, candidate paths are determined as possible controlled evolutions which fulfill the given specifications. A validation scheme comprising three different techniques is applied to determine whether the candidate path represents a feasible control strategy for the hybrid system. If the specification is violated, the abstract model is refined according to the validation result. The iterative application of the determination of candidate paths, the path validation and the model refinement steers the search for a control strategy such that often only a relatively small part of the reachable state space has to be explored. The synthesis procedure is illustrated for two examples.     

Controllability and stabilizability of linear time‐varying distributed hereditary control systems

This paper is concerned with the controllability and stabilizability problem for control systems described by a time‐varying linear abstract differential equation with distributed delay in the state variables. An approximate controllability property is established, and for periodic systems, the stabilization problem is studied. Assuming that the semigroup of operators associated with the uncontrolled and non delayed equation is compact, and using the characterization of the asymptotic stability in terms of the spectrum of the monodromy operator of the uncontrolled system, it is shown that the approximate controllability property is a sufficient condition for the existence of a periodic feedback control law that stabilizes the system. The result is extended to include some systems which are asymptotically periodic. Copyright © 2014 John Wiley & Sons, Ltd.     

Solution theory for systems of bilinear equations

Bilinear systems of equations are defined, motivated and analysed for solvability. Elementary structure is mentioned and it is shown that all solutions may be obtained as rank one completions of a linear matrix polynomial derived from elementary operations. This idea is used to identify bilinear systems that are solvable for all right-hand sides and to understand solvability when the number of equations is large or small.     

Argumentation update in YALLA (Yet Another Logic Language for Argumentation)

This article proposes a complete framework for handling the dynamics of an abstract argumentation system. This frame can encompass several belief bases under the form of several argumentation systems, more precisely it is possible to express and study how an agent who has her own argumentation system can interact on a target argumentation system (that may represent a state of knowledge at a given stage of a debate). The two argumentation systems are defined inside a reference argumentation system called the universe which constitutes a kind of “common language”. This paper establishes three main results. First, we show that change in argumentation in such a framework can be seen as a particular case of belief update. Second, we have introduced a new logical language called YALLA in which the structure of an argumentation system can be encoded, enabling to express all the basic notions of argumentation theory (defense, conflict-freeness, extensions) by formulae of YALLA. Third, due to previous works about dynamics in argumentation we have been in position to provide a set of new properties that are specific for argumentation update.     

Discrete-time reduced-order estimator design for bilinear stochastic systems with error covariance assignment

This paper is concerned with the design of reduced-order stateestimators for bilinear stochastic discrete-time systems subjectedto estimation error covariance assignment. The purpose of theproblem addressed is to design the reduced-order state estimators for the bilinear stochastic discrete-time systems such thatthe steady-state estimation error covariances achieve the prespecified values. A simple, effective matrix inequality approach is developed to solve this problem. Specifically, (1) the parameterisationof estimation error covariances that certain bilinear errordynamic processes may possess is presented, (2) the characterisationof all reduced-order state estimators that assign such errorcovariances is explicitly derived, and (3) the solvabilityof the assignability conditions is discussed. Furthermore,an illustrative example is used to demonstrate the effectivenessof the proposed design procedure.     

Controllability of Some Nonlinear Systems in Hilbert Spaces

In this paper, several abstract results concerning the controllability of semilinear evolution systems are obtained. First, approximate controllability conditions for semilinear systems are obtained by means of a fixed-point theorem of the Rothe type; in this case, the compactness of the linear operator is assumed. Next, the exact controllability of semilinear systems with nonlinearities having small Lipschitz constants is derived by means of the Banach fixed-point theorem; in this case, the compactness of the operators is not assumed. In both cases, it is proven that the controllability of the linear system implies the controllability of the associated semilinear system. Finally, these abstract results are applied to the controllability of the semilinear wave and heat equations.     

增生映射的扰动理论对与广义(p,q)-Laplace算子相关的非线性椭圆系的应用

Using perturbation results on the sums of ranges of nonlinear accretive mappings of Calvert and Gupta, we present some abstract results for the existence of the solutions of nonlinear Neumann elliptic systems which is related to the so-called generalized （p, q）-Laplacian in this paper. The systems discussed in this paper and the method used extend and complement some of the previous work.     

Improved results on fixed-time stability of switched nonlinear time-varying systems

This paper considers the problem of fixed-time stability (FTS) for switched nonlinear time-varying (NTV) systems. Firstly, three sufficient conditions are proposed to verify the FTS of NTV systems by using the improved Lyapunov function, which has a tighter upper bound of time derivative. Then, two FTS conditions are given for the switched NTV system by extending the obtained results, moreover, a switching strategy is also provided by using the minimum dwell time method. Finally, the obtained results are extended to study the FTS of impulsive NTV systems. Comparing with the existing results, the obtained conditions have two improvements: (1) provides a more accurate estimate for the upper bound of settling time of NTV systems, and (2) allows the Lyapunov function to increase at the switching instant of switched NTV (or impulsive NTV) systems. Two numerical examples are given to illustrate the theoretical results.     

Stability of bilinear time-delay systems

In this paper, the stability of the differential bilinear time-delaysystems is first studied. We consider time-varying bilineartime-delay systems with output feedback. The input or controlu(t)is not only a signal but also an input with output feedback.The analysis is given by using norm-transformation methods.     

Computations via Newtonian and relativistic kinematic systems

We are developing a rigorous methodology to analyse experimental computation, by which we mean the idea of computing a set or function by experimenting with some physical equipment. Here we consider experimental computation by kinematic systems under both Newtonian and relativistic kinematics. An experimental procedure, expressed in a language similar to imperative programming languages, is applied to equipment, having the form of a bagatelle, and is interpreted using the two theories. We prove that for any set A of natural numbers there exists a two-dimensional kinematic system B_A with a single particle P whose observable behaviour decides n∈A for all n∈N. The procedure can operate under (a) Newtonian mechanics or (b) relativistic mechanics. The proofs show how any information (coded by some A) can be embedded in the structure of a simple kinematic system and retrieved by simple observations of its behaviour. We reflect on the methodology, which seeks a formal theory for performing abstract experiments with physical restrictions on the construction of systems. We conclude with some open problems.     

Some classes of abstract simplicial complexes motivated by module theory

In this paper we analyze some classes of abstract simplicial complexes relying on algebraic models arising from module theory. To this regard, we consider a left-module on a unitary ring and find models of abstract complexes and related set operators having specific regularity properties, which are strictly interrelated to the algebraic properties of both the module and the ring.Next, taking inspiration from the aforementioned models, we carry out our analysis from modules to arbitrary sets. In such a more general perspective, we start with an abstract simplicial complex and an associated set operator. Endowing such a set operator with the corresponding properties obtained in our module instances, we investigate in detail and prove several properties of three subclasses of abstract complexes.More specifically, we provide uniformity conditions in relation to the cardinality of the maximal members of such classes. By means of the notion of OSS-bijection, we prove a correspondence theorem between a subclass of closure operators and one of the aforementioned families of abstract complexes, which is similar to the classic correspondence theorem between closure operators and Moore systems. Next, we show an extension property of a binary relation induced by set systems when they belong to one of the above families.Finally, we provide a representation result in terms of pairings between sets for one of the three classes of abstract simplicial complexes studied in this work.     

Geometric Aspects of Force Controllability for a Swimming Model

We study controllability properties (swimming capabilities) of a mathematical model of an abstract object which “swims” in the 2-D Stokes fluid. Our goal is to investigate how the geometric shape of this object affects the forces acting upon it. Such problems are of interest in biology and engineering applications dealing with propulsion systems in fluids. This work was supported in part by NSF Grant DMS-0504093.