Convergence Analysis of the Implicit Euler-discretization and Sufficient Conditions for Optimal Control Problems Subject to Index-one Differential-algebraic Equations

Martens, Björn; Gerdts, Matthias

doi:10.1007/s11228-018-0471-x

Cited by 6 publications

(1 citation statement)

References 18 publications

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“…The boundary conditions are given as follows:

where Ω ⊂ R 2 , is the polygonal region whose boundary is marked as ∂ Ω [ 14 , 15 ]. Where v is the normal vector outside the unit of ∂ Ω, J =(0, T] and K are the tensor of the square integrable symmetric positive definite.…”

Section: Multivariable Linear Algebraic Discrete Methods For Npesmentioning

confidence: 99%

Multivariable Linear Algebraic Discretization of Nonlinear Parabolic Equations for Computational Analysis

Zuo¹,

Mei²

2022

Computational Intelligence and Neuroscience

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Since the nonlinear parabolic equation has many variables, its calculation process is mostly an algebraic operation, which makes it difficult to express the discrete process concisely, which makes it difficult to effectively solve the two grid algorithm problems and the convergence problem of reaction diffusion. The extended mixed finite element method is a common method for solving reaction-diffusion equations. By introducing intermediate variables, the discretized algebraic equations have great nonlinearity. To address this issue, the paper proposes a multivariable linear algebraic discretization method for NPEs. First, the NPE is discretized, the algebraic form of the nonlinear equation is transformed into the vector form, and the rough set (RS) and information entropy (IE) are constructed to allocate the weights of different variable attributes. According to the given variable attribute weight, the multiple variables in the equation are discretized by linear algebra. It can effectively solve the two grid algorithm problems and the convergence problem of reaction diffusion and has good adaptability in this field.

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“…The boundary conditions are given as follows:

Section: Multivariable Linear Algebraic Discrete Methods For Npesmentioning

confidence: 99%

Multivariable Linear Algebraic Discretization of Nonlinear Parabolic Equations for Computational Analysis

Zuo¹,

Mei²

2022

Computational Intelligence and Neuroscience

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Error Analysis for the Implicit Euler Discretization of Linear-Quadratic Control Problems with Higher Index DAEs and Bang–Bang Solutions

Martens

Gerdts

2020

Differential-Algebraic Equations Forum

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Error estimates for Runge–Kutta schemes of optimal control problems with index 1 DAEs

Martens

2023

Comput Optim Appl

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In this paper we derive error estimates for Runge–Kutta schemes of optimal control problems subject to index one differential–algebraic equations (DAEs). Usually, Runge–Kutta methods applied to DAEs approximate the differential and algebraic state in an analogous manner. These schemes can be considered as discretizations of the index reduced system where the algebraic equation is solved for the algebraic variable to get an explicit ordinary differential equation. However, in optimal control this approach yields discrete necessary conditions that are not consistent with the continuous necessary conditions which are essential for deriving error estimates. Therefore, we suggest to treat the algebraic variable like a control, obtaining a new type of Runge–Kutta scheme. For this method we derive consistent necessary conditions and compare the discrete and continuous systems to get error estimates up to order three for the states and control as well as the multipliers.

show abstract

Convergence Analysis of the Implicit Euler-discretization and Sufficient Conditions for Optimal Control Problems Subject to Index-one Differential-algebraic Equations

Cited by 6 publications

References 18 publications

Multivariable Linear Algebraic Discretization of Nonlinear Parabolic Equations for Computational Analysis

Multivariable Linear Algebraic Discretization of Nonlinear Parabolic Equations for Computational Analysis

Error Analysis for the Implicit Euler Discretization of Linear-Quadratic Control Problems with Higher Index DAEs and Bang–Bang Solutions

Error estimates for Runge–Kutta schemes of optimal control problems with index 1 DAEs

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