Numerical Methods for Delay Differential Equations

Bellen, Alfredo; Zennaro, Marino

doi:10.1093/acprof:oso/9780198506546.001.0001

Cited by 637 publications

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“…of (1.1) on certain equidistant step-values {t n (= nh)} with the step-size h. We introduce the following new definition of delay-dependent stability of numerical methods for system (1.1). Remark 4.1 In [2,7,18], the definition of delay-dependent stability, which is called D-stability there, is too restricted since it requires for all asymptotically stable delay differential system (1.3) and for all the natural numbers m the resulting numerical solution is asymptotically stable. Therefore, almost all the standard RK methods are excluded in the D-stability sense.…”

Section: Delay-dependent Stability Of Numerical Methodsmentioning

confidence: 99%

“…The stability which does not depend on delays is called delay-independent, otherwise it is referred to as delay-dependent (they are discussed in [1,2,4,[9][10][11]19]). Each case is extended to the neutral type and the reader can refer to [2,10] for the delay-independent and to [2,9,11,19] for the delaydependent case. Then, the stability of numerical methods is also divided into delayindependent and delay-dependent according to which system the method is applied to.…”

Section: Introductionmentioning

confidence: 99%

“…Although numerical delay-independent stability has been discussed in [1,2,10], only a few works have been reported for the delay-dependent case [2,7,13,18]. The literature [2,7,18] proposed the delay-dependent stability of numerical methods for the systemu (t) = Lu(t) + Mu(t − τ ), (1.3) and called it as D-stability.…”

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

“…The literature [2,7,18] proposed the delay-dependent stability of numerical methods for the systemu (t) = Lu(t) + Mu(t − τ ), (1.3) and called it as D-stability. However, as pointed out in [2], delay-dependent stability of numerical methods for the system is a real challenge. In fact, in [18] it is proved that for any A-stable natural Runge-Kutta method for system (1.3) and for any step-size h = τ/m (m is a positive integer), there exists an asymptotically stable linear system of dimension d = 4 for which u n → 0 as n → ∞ does not hold.…”

Section: Introductionmentioning

confidence: 99%

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Delay-dependent stability of numerical methods for delay differential systems of neutral type

Mitsui

2017

Bit Numer Math

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We are concerned with stability of numerical methods for delay differential systems of neutral type. In particular, delay-dependent stability of numerical methods is investigated. By means of the H -matrix norm, a necessary and sufficient condition for the asymptotic stability of analytic solution of linear neutral differential systems is derived. Then, based on the argument principle, sufficient conditions for delay-dependent stability of Runge-Kutta and linear multi-step methods are presented, respectively. Furthermore, two algorithms are provided for checking delay-dependent stability of analytical and numerical solutions, respectively. Numerical examples are given to illustrate the main results.

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Section: Delay-dependent Stability Of Numerical Methodsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Delay-dependent stability of numerical methods for delay differential systems of neutral type

Mitsui

2017

Bit Numer Math

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Economic model predictive control of nonlinear time‐delay systems: Closed‐loop stability and delay compensation

Ellis

Christofides

2015

AIChE Journal

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in Wiley Online Library (wileyonlinelibrary.com) Closed-loop stability of nonlinear time-delay systems under Lyapunov-based economic model predictive control (LEMPC) is considered. LEMPC is initially formulated with an ordinary differential equation model and is designed on the basis of an explicit stabilizing control law. To address closed-loop stability under LEMPC, first, we consider the stability properties of the sampled-data system resulting from the nonlinear continuous-time delay system with state and input delay under a sample-and-hold implementation of the explicit controller. The steady-state of this sampled-data closed-loop system is shown to be practically stable. Second, conditions such that closed-loop stability, in the sense of boundedness of the closedloop state, under LEMPC are derived. A chemical process example is used to demonstrate that indeed closed-loop stability is maintained under LEMPC for sufficiently small time-delays. To cope with performance degradation owing to the effect of input delay, a predictor feedback LEMPC methodology is also proposed. The predictor feedback LEMPC design employs a predictor to compute a prediction of the state after the input delay period and an LEMPC scheme that is formulated with a differential difference equation (DDE) model, which describes the time-delay system, initialized with the predicted state. The predictor feedback LEMPC is also applied to the chemical process example and yields improved closed-loop stability and economic performance properties. V C 2015 American Institute of Chemical Engineers AIChE J, 61: 4152-4165, 2015 Keywords: economic model predictive control, nonlinear time-delay systems, process control, process economics, process optimization IntroductionWithin the context of control of chemical process systems, time-delays resulting from computation and communication delays are enviable albeit these time-delays may be small and insignificant depending on the magnitude of the time-delay relative to the time-constants of the process dynamics. From a modeling perspective, time-delays are often employed to describe and/or to approximate the dynamics of transportation of material through the process system (e.g., flow through pipes), control actuator dynamics, measurement sensor dynamics, and high-order dynamic behavior. Thus, robustness with respect to closed-loop stability and performance of control systems to time-delays is an important consideration. Moreover, many chemical process systems have significant nonlinear behavior owing to complex reaction mechanisms, Arrhenius reaction-rate dependence on temperature, and thermodynamic relationships which adds additional complexity in considering the closed-loop behavior of the system resulting from a nonlinear time-delay system under a control law.Dynamic models of systems that involve nonlinearities and time-delays are systems of nonlinear differential difference equations (DDEs). Systems described by DDEs are fundamentally different than systems described by ordinary differential equat...

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Extended cubic B‐spline collocation method for singularly perturbed parabolic differential‐difference equation arising in computational neuroscience

Daba

Duressa

2020

Numer Methods Biomed Eng

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A parameter uniform numerical method is presented for solving singularly perturbed parabolic differential‐difference equations with small shift arguments in the reaction terms arising in computational neuroscience. To approximate the terms with the shift arguments, Taylor's series expansion is used. The resulting singularly perturbed parabolic differential equation is solved by applying the implicit Euler method in temporal direction and extended cubic B‐spline basis functions consisting of a free parameter λ for the resulting system of ordinary differential equations in the spatial direction. The proposed method is shown to be accurate of order O()normalΔt+h2ε+h by preserving an ε− uniform convergence. To demonstrate the applicability of the proposed method, two test examples are solved by the method and the numerical results are compared with some existing results. The obtained numerical results agreed with the theoretical results.

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Numerical Methods for Delay Differential Equations

Cited by 637 publications

References 0 publications

Delay-dependent stability of numerical methods for delay differential systems of neutral type

Delay-dependent stability of numerical methods for delay differential systems of neutral type

Economic model predictive control of nonlinear time‐delay systems: Closed‐loop stability and delay compensation

Extended cubic B‐spline collocation method for singularly perturbed parabolic differential‐difference equation arising in computational neuroscience

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