Linear differential algebraic equations of index 1 and their adjoint equations

Balla, K.; März, Roswitha

doi:10.1007/bf03322509

Cited by 20 publications

(17 citation statements)

References 6 publications

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“…As in the case of ODEs it is useful to introduce the adjoint equation to (9), see also [14,5,51,52].…”

Section: Lyapunov Exponents and Lyapunov Spectral Intervalsmentioning

confidence: 99%

“…That is, given a DAE, it may happen that its adjoint DAE does not exist or sometimes it is not clear at all what is an adjoint system. For more details on adjoint DAEs, see [5,14] and references therein.…”

Section: Remark 11mentioning

confidence: 99%

“…The identities 1.-2. in Proposition 43 mean that X − (t 1 , t 0 ) is a reflexive generalized inverse of X(t 1 , t 0 ), while the identities 3.-4. guarantee that this generalized inverse is unique, see [5,6].…”

Section: E(t)ẋ(t) = A(t)x(t) T ∈ Imentioning

confidence: 99%

“…Here we allow general regular DAEs of arbitrary index and we use reformulations based on derivative arrays as well as the strangeness index concept [50]. As in the ODE case there is also a close relation of the spectral theory to the theory of adjoint equations which has recently been studied in the context of control problems in [4,5,6,14,51,52].…”

Section: Introductionmentioning

confidence: 99%

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Lyapunov, Bohl and Sacker-Sell Spectral Intervals for Differential-Algebraic Equations

Linh

Mehrmann

2009

J Dyn Diff Equat

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Lyapunov and exponential dichotomy spectral theory is extended from ordinary differential equations (ODEs) to nonautonomous differential-algebraic equations (DAEs). By using orthogonal changes of variables, the original DAE system is transformed into appropriate condensed forms, for which concepts such as Lyapunov exponents, Bohl exponents, exponential dichotomy and spectral intervals of various kinds can be analyzed via the resulting underlying ODE. Some essential differences between the spectral theory for ODEs and that for DAEs are pointed out. It is also discussed how numerical methods for computing the spectral intervals associated with Lyapunov and Sacker-Sell (exponential dichotomy) can be extended from those methods proposed for ODEs. Some numerical examples are presented to illustrate the theoretical results.

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“…As in the case of ODEs it is useful to introduce the adjoint equation to (9), see also [14,5,51,52].…”

Section: Lyapunov Exponents and Lyapunov Spectral Intervalsmentioning

confidence: 99%

Section: Remark 11mentioning

confidence: 99%

Section: E(t)ẋ(t) = A(t)x(t) T ∈ Imentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Lyapunov, Bohl and Sacker-Sell Spectral Intervals for Differential-Algebraic Equations

Linh

Mehrmann

2009

J Dyn Diff Equat

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“…Equation (10) will enable us to get rid of thex, which does not need to be calculated explicitly. We choose λ(t) ∈ R N ×F appropriately and require that λ(t) satisfies the linear 'adjoint' DAE (we assume the index 1 case, which is not trivial [4]). …”

Section: Transient Sensitivity Analysismentioning

confidence: 99%

Adjoint Transient Sensitivity Analysis in Circuit Simulation

Ilievski

Xu²,

Verhoeven

et al. 2007

Scientific Computing in Electrical Engineering

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Summary. Sensitivity analysis is an important tool that can be used to assess and improve the design and accuracy of a model describing an electronic circuit. Given a model description in the form of a set of differential-algebraic equations it is possible to observe how a circuit's output reacts to varying input parameters, which are introduced at the requirements stage of design. In this paper we consider the adjoint method more closely. This method is efficient when the number of parameters is large. We extend the transient sensitivity work of Petzold et al., in particular we take into account the parameter dependency of the dynamic term. We also compare the complexity of the direct and adjoint sensitivity and derive some error estimates. Finally we sketch out how Model Order Reduction techniques could be used to improve the efficiency of adjoint sensitivity analysis.

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