Topic
Piecewise
About: Piecewise is a research topic. Over the lifetime, 21064 publications have been published within this topic receiving 432096 citations. The topic is also known as: piecewise-defined function & hybrid function.
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TL;DR: The Lyapunov function, in quadratic form, is assigned to each subsystem such that it is non-increasing at the switching instants, and is used in deriving state-feedback control law that robustly achieves a prescribed L2 -gain bound.
Abstract: A state-dependent switching law that obeys a dwell time constraint and guarantees the stability of a switched linear system is designed. Sufficient conditions are obtained for the stability of the switched systems when the switching law is applied in presence of polytopic type parameter uncertainty. A Lyapunov function, in quadratic form, is assigned to each subsystem such that it is non-increasing at the switching instants. During the dwell time, this function varies piecewise linearly in time. After the dwell, the system switches if the switching results in a decrease in the value of the LF. The method proposed is also applicable to robust stabilization via state-feedback. It is further extended to guarantee a bound on the L2-gain of the switching system; it is also used in deriving state-feedback control law that robustly achieves a prescribed L2 -gain bound.
194 citations
TL;DR: In this paper, it was shown that invariant measures depend continuously on three types of perturbations: deterministic perturbation, stochastic perturbance, and randomly occuring deterministically occuring perturbant.
Abstract: For a certain class of piecewise monotonic transformations it is shown using a spectral decomposition of the Perron-Frobenius-operator ofT that invariant measures depend continuously on 3 types of perturbations: 1) deterministic perturbations, 2) stochastic perturbations, 3) randomly occuring deterministic perturbations. The topology on the space of perturbed transformations is derived from a metric on the space of Perron-Frobenius-operators.
194 citations
TL;DR: It will be shown by a series of numerical experiments that the algorithm can solve the problem of practical size in an efficient manner.
Abstract: We will propose a branch and bound algorithm for calculating a globally optimal solution of a portfolio construction/rebalancing problem under concave transaction costs and minimal transaction unit constraints. We will employ the absolute deviation of the rate of return of the portfolio as the measure of risk and solve linear programming subproblems by introducing (piecewise) linear underestimating function for concave transaction cost functions. It will be shown by a series of numerical experiments that the algorithm can solve the problem of practical size in an efficient manner.
193 citations
TL;DR: By combining a novel Lyapunov-Krasovskii functional, which benefits from the delay partitioning method, with the free-weighting matrix technique, sufficient conditions are proposed to guarantee the exponential stability for the switched neural networks with constant and time-varying delays, respectively.
Abstract: This paper is concerned with the problem of exponential stability analysis of continuous-time switched delayed neural networks. By using the average dwell time approach together with the piecewise Lyapunov function technique and by combining a novel Lyapunov-Krasovskii functional, which benefits from the delay partitioning method, with the free-weighting matrix technique, sufficient conditions are proposed to guarantee the exponential stability for the switched neural networks with constant and time-varying delays, respectively. Moreover, the decay estimates are explicitly given. The results reported in this paper not only depend upon the delay but also depend upon the partitioning, which aims at reducing the conservatism. Numerical examples are presented to demonstrate the usefulness of the derived theoretical results.
193 citations
TL;DR: A nonoverlapping variant of the Schwarz waveform relaxation algorithm for wave propagation problems with variable coefficients in one spatial dimension is introduced, and convergence in two iterations can be achieved independently of the number of subdomains in that case.
Abstract: We introduce a nonoverlapping variant of the Schwarz waveform relaxation algorithm for wave propagation problems with variable coefficients in one spatial dimension. We derive transmission conditions which lead to convergence of the algorithm in a number of iterations equal to the number of subdomains, independently of the length of the time interval. These optimal transmission conditions are in general nonlocal, but we show that the nonlocality depends on the time interval under consideration, and we introduce time windows to obtain optimal performance of the algorithm with local transmission conditions in the case of piecewise constant wave speed. We show that convergence in two iterations can be achieved independently of the number of subdomains in that case. The algorithm thus scales optimally with the number of subdomains, provided the time windows are chosen appropriately. For continuously varying coefficients we prove convergence of the algorithm with local transmission conditions using energy estimates. We then introduce a finite volume discretization which permits computations on nonmatching grids, and we prove convergence of the fully discrete Schwarz waveform relaxation algorithm. We finally illustrate our analysis with numerical experiments.
193 citations