Discrete-time switched descriptor systems: How to solve them?

This paper deals with interval state estimation for continuous-time linear parameter varying (LPV) switched systems with measured polytopic parameter dependence, assuming a known switching signal. Considering that the measurement noise and the state disturbance are unknown but bounded and that the dynamics of the system are described by a convex combination, lower and upper bounds of the state are therefore determined. An interval observer is designed in order to guarantee both stability and cooperativity of the upper and lower observation errors. For the proposed observer, sufficient input-to-state stability conditions are given in terms of linear matrix inequalities, adopting firstly common quadratic Lyapunov functions, under arbitrary switching signals and secondly multiple quadratic Lyapunov functions, under average dwell time (ADT) switching signals. The efficiency of the proposed interval observer is highlighted through simulation results on academic examples.

ABSTRACTThis paper introduces a new class of discrete-time two-dimensional (2-D) switched systems with multiplicative noise. Firstly, we extend the definition of the asymptotic stability and establish a sufficient stochastically stability condition for this new model under arbitrary switching signal. Then, by introducing the average dwell time into this new model and combining with the Lyapunov function, we investigate the extended stochastic exponential stability of the 2-D switched systems with multiplicative noise for the restricted switching case. Moreover, some remarks and discussions are given to illustrate the significance of obtained results, which generalise and comprise some previous results the literature. Finally, two examples are provided to show the effectiveness of the theoretical results.

Stabilizability analysis and switching signals design of switched Boolean networks

This paper discusses the existence and uniqueness of strong solutions for the first order homogeneous linear generalized distributed parameter system and nonhomogeneous linear generalized distributed parameter system by using the generalized operator semigroup theory. And sufficient conditions of existence and uniqueness of strong solutions and constructive expressions of strong solutions are given in the paper. Further, the existence of strong solutions of semilinear generalized distributed parameter system is studied by mild solutions.

In this article, we consider a nonlinear Sobolev type fractional functional integrodifferential equations in a Banach space along with a nonlocal condition. Sufficient conditions for existence, uniqueness and dependence on initial data of local solutions of considered problem are derived by employing fixed point techniques and theory of classical semigroup. Further, we also render the criteria for existence of global solution. At the end, we provide an application to elaborate the obtained results.

Modelling the temporal response of travellers to transport policy interventions has rapidly emerged as a major issue in many practical transport planning studies and is recognised to hold particular challenges. The importance of congestion and its variation over the day, together with the emergence of time-dependent road user charging as a policy tool, emphasise the need to understand whether and how travellers will change the timing of their journeys. For practical planning studies, analysts face a major issue of relating temporal changes to other behavioural changes that are likely to result from policy or exogenous changes. In particular, the relative sensitivity of time and mode switching has been difficult to resolve. This paper describes a study undertaken to determine the relative sensitivity of mode and time of day choice to changes in travel times and costs and to investigate whether evidence exists of varying magnitudes of unobservable influences in time of day switching. The study draws on data from three related stated preference studies undertaken over the past decade in the United Kingdom and the Netherlands and uses error components logit models to investigate the patterns of substitution between mode and time of day alternatives. It is concluded that the magnitude of unobserved influences on time switching depends significantly on the magnitudes of the time switches considered. With time periods of the magnitude generally represented in practical modelling, i.e. peak periods of 2–3 hours, time switching is generally more sensitive in these data than mode switching. However, the context of the modelling and the extent to which relevant variables can be measured will strongly influence these results.

Infinite Coupled Riccati Equations Arising in Certain Optimal Control Problem

The solvability of a delay differential equation arising in the construction of quadratic cost functionals, i.e. Lyapunov functionals, for a linear time-delay system with a constant and a distributed delay is investigated. We present a delay-free auxiliary ordinary differential equation system with algebraically coupled split-boundary conditions, that characterizes the solutions of the delay differential equation and is used for solution synthesis. A spectral property of the time-delay system yields a necessary and sufficient condition for existence and uniqueness of solutions to the auxiliary system, equivalently the delay differential equation. The result is a tractable analytic solution framework to the delay differential equation.

We consider parabolic variational inequalities in Sobolev spaces and generalized Lebesgue spaces. Sufficient conditions of existence and uniqueness of solution of these inequalities have been established. We have also proved some comparison theorems.

We present a unified approach to study large positive solutions (i.e., u ( x ) → ∞ u(x)\to \infty as x → ∂ Ω x\to \partial \Omega ) of the equation Δ u + h u − k ψ ( u ) = − f \Delta u+hu-k\psi (u)=-f in an arbitrary domain Ω \Omega . We assume ψ ( u ) \psi (u) is convex and grows sufficiently fast as u → ∞ u\to \infty . Equations of this type arise in geometry (Yamabe problem, two dimensional curvature equation) and probability (superdiffusion). We prove that both existence and uniqueness are local properties of points of the boundary ∂ Ω \partial \Omega ; i.e., they depend only on properties of Ω \Omega in arbitrarily small neighborhoods of each boundary point. We also find several new necessary and sufficient conditions for existence and uniqueness of large solutions including an existence theorem on domains with fractal boundaries.

A steepest descent method is constructed for the general setting of a linear differential equation paired with uniqueness-inducing conditions which might yield a generally overdetermined system. The method differs from traditional steepest descent methods by considering the conditions when defining the corresponding Sobolev space. The descent method converges to the unique solution to the differential equation so that change in condition values is minimal. The system has a solution if and only if the first iteration of steepest descent satisfies the system. The finite analogue of the descent method is applied to example problems involving finite difference equations. The well-posed problems include a singular ordinary differential equation and Laplace’s equation, each paired with respective Dirichlet-type conditions. The overdetermined problems include a first-order nonsingular ordinary differential equation with Dirichlet-type conditions and the wave equation with both Dirichlet and Neumann conditions. The method is applied in an investigation of the Tricomi equation, a long-studied equation which acts as a prototype of mixed partial differential equations and has application in transonic flow. The Tricomi equation has been studied for at least ninety years, yet necessary and sufficient conditions for existence and uniqueness of solutions on an arbitrary mixed domain remain unknown. The domains of interest are rectangular mixed domains. A new type of conditions is introduced. Ladder conditions take the uncommon approach of specifying information on the interior of a mixed domain. Specifically, function values are specified on the parabolic portion of a mixed domain. The remaining conditions are specified on the boundary. A conjecture is posed and states that ladder conditions are necessary and sufficient for existence and uniqueness of a solution to the Tricomi equation. Numerical experiments, produced by application of the descent method, provide strong evidence in support of the conjecture. Ladder conditions allow for a continuous deformation from Dirichlet conditions to initial-boundary value conditions. Such a deformation is applied to a class of Tricomi-type equations which transition from degenerate elliptic to degenerate hyperbolic. A conjecture is posed and states that each problem is uniquely solvable and the solutions vary continuously as the differential equation and corresponding conditions vary continuously. If the conjecture holds true, the result will provide a method of unifying elliptic Dirichlet problems and hyperbolic initial-boundary value problem. Numerical evidence in support of the conjecture is presented.

SummaryThis paper presents an approach to the stability analysis of a class of nonlinear interconnected continuous‐time singular systems with arbitrary switching signals. This class of interconnected subsystems consists of unknown but bounded state delay and nonlinear terms, and each subsystem can be globally stable, unstable, or locally stable. By constructing a new Lyapunov‐like Krasovskii functional, sufficient conditions are derived and formulated to check the asymptotic (exponential) stability of such systems with arbitrary switching signals. Then, some new general criteria for asymptotic (exponential) stability with average dwell‐time switching signals are also established. The theoretical developments are demonstrated by two numerical simulations. Copyright © 2014 John Wiley & Sons, Ltd.

Pinning bipartite synchronization for coupled reaction–diffusion neural networks with antagonistic interactions and switching topologies

Stability and [formula omitted]-gain analysis for switched delay systems: A delay-dependent method

Building on our recently proposed model reduction methods for switched ordinary linear systems, we propose a comprehensive model reduction method for linear switched differential-algebraic equations (DAEs). In contrast to most other available model reduction methods for switched systems, we consider the switching signal as a given time-variance of the system. This allows us to exploit certain linear subspaces in the reduction process and also provide in general significantly smaller reduced models compared to methods which consider arbitrary switching signals. Model reduction for switched DAEs has some unique features that make a generalization of the available methods nontrivial; in particular, the presence of jumps and Dirac impulses in response to switches have to be carefully treated. Furthermore, due to the algebraic constraints, the reachability subspaces cannot be the full space, hence a straightforward application of balanced truncation is not possible (because the corresponding reachability Gramians will be structurally non-invertible). We resolve this problem by first applying an exact model reduction which reduces the switched DAE to a switched ordinary system with jumps and which carefully keeps track of the impulsive effects. As a second step, we then apply a midpoint balanced truncation approach to further reduce the switched system. In addition to the challenge to appropriately take into account the Dirac impulses, another novel challenge was the occurrence of input-dependent state-jumps. We propose to deal with input-dependent jumps by combining certain discrete-time reachability Gramians with continuous-time reachability Gramians. We provide corresponding Matlab implementations of the proposed algorithms and illustrate their effectiveness with some academic examples.