The parameterization method in optimal control problems and differential–algebraic equations

Gorbunov, V. K.; Lutoshkin, Igor

doi:10.1016/j.cam.2005.03.017

Cited by 11 publications

(3 citation statements)

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“…To simplify we will consider a special method to reduce the index by one which will result in an index-l DAE, see Babolian and Hosseini [5]. The methods of reducing and modified reducing index have be generalized and numerically implemented by Hosseini [14], the parameterization method for DAE arising in optimal control problems was done by Gorbunov and Lutoshkin [13] while fractional step methods were considered by Vijalapura et al [19]. More on such index-l DAE can be found in Attili[l,2] where the systems considered have some types of singularities.…”

Section: Introductionmentioning

confidence: 99%

The Differential Transform Method for the Numerical Treatment of Differential-Algebraic Equations with Index 2

Attili

Tabash

Dhaheri

et al. 2009

Journal of Dynamical Systems and Geometric Theories

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We will consider the numerical solution of the Hessenberg linear index-2 differential algebraic equations (DAE's) using the differential transform method. We Will first present a method to reduce the mdex of the DAE to index-I. This results in a problem easier to solve. Numerical examples in additIOn to the comparison With the work of others will also be presented to show the effiCiency of the method.

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Section: Introductionmentioning

confidence: 99%

The Differential Transform Method for the Numerical Treatment of Differential-Algebraic Equations with Index 2

Attili

Tabash

Dhaheri

et al. 2009

Journal of Dynamical Systems and Geometric Theories

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“…The whole time domain of a continuous admissible input is partitioned into several subintervals, and the input for each subinterval is approximated by a piecewise constant function. The dynamic log gains with respect to the continuous admissible input can be computed based on the partial derivations of dependent variables with respect to the piecewise constant input [ 28 - 30 ].…”

Section: Introductionmentioning

confidence: 99%

Dynamic sensitivity analysis of biological systems

Wang

Chang

2008

BMC Bioinformatics

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Background: A mathematical model to understand, predict, control, or even design a real biological system is a central theme in systems biology. A dynamic biological system is always modeled as a nonlinear ordinary differential equation (ODE) system. How to simulate the dynamic behavior and dynamic parameter sensitivities of systems described by ODEs efficiently and accurately is a critical job. In many practical applications, e.g., the fed-batch fermentation systems, the system admissible input (corresponding to independent variables of the system) can be timedependent. The main difficulty for investigating the dynamic log gains of these systems is the infinite dimension due to the time-dependent input. The classical dynamic sensitivity analysis does not take into account this case for the dynamic log gains.

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“…The derivatives with respect to the control parameters may be effectively calculated with the help of variational techniques and adjoint variables. The PM appeared very effective for different degenerate OC problems, and it was expanded to classical calculus of variation (CV) problems which arise in connection with DAEs, especially in cases of their essential degeneracy [5], [6].…”

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confidence: 99%

Variational splines for the numerical solution of singular differential equations

Gorbunov

Lutoshkin

Martynenko

2007

Proc Appl Math and Mech

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The numerical parametrization method (PM), originally created for optimal control problems, is specificated for classical calculus of variation problems that arise in connection with singular implicit (IDEs) and differential-algebraic equations (DAEs). The PM for IDEs is based on representation of the required solution as a spline with moving knots and on minimization of the discrepancy functional with respect to the spline parameters. Such splines are named variational splines. For DAEs only finite entering functions can be represented by splines, and the functional under minimization is the discrepancy of the algebraic subsystem. The first and the second derivatives of the functionals are calculated in two ways -for DAEs with the help of adjoint variables, and for IDE directly. The PM does not use the notion of differentiation index, and it is applicable to any singular equation having a solution.The last three decades mathematical modeling involves problems that are modeled as systems of implicit differential equationswhereSuch systems arise in designing electric circuits (microchips for computers), mechanics (cinematics equations), nuclear energy, and others [8]. Irresolvability of the system (1) with respect to the derivativesẋ (singularity) can be a consequence of the complexity of the corresponding algebraic and transcendental transformations, as well as a consequence of the degeneration of the Jacobi matrix ∂F (ẋ, x, t)/∂ẋ on the solution {x(t)}. The statement of initial or boundary value problems for such equations should be in accordance with the system (1). The IDEs (1) are also called 'unstructured differential-algebraic equations' [1].An important particular kind of IDEs is the so-called 'structured DAEs', or simply DAEṡwhereHere initial or boundary value problems should be in accordance with the finite subsystem, and difficulties of theory and numerical solution to the initial/boundary value problems appear when the Jacobi matrix of the finite subsystem of ∂g(x, u, t)/∂u degenerates on the solution {x(t)} or {x(t), u(t)}.It is known that singular differential equations has a set of pathologies with respect to regular ODEs. So, their solutions (if they exist) can depend non-continuously on input data (initial values and parameters of equations), and the set of solutions can be infinite-dimensional (see, for example, the recent investigation [2]). Correspondingly, in the common singular case, theory and numerical methods should be created in frame of the theory of ill-posed problems [4].Often the system IDE or DAE can be transformed to the regular normal (Cauchy) by a finite number of differentiations and by algebraic transformations. The minimal number of such differentiations is called the differentiation index (DI) [8]. In applications, DAEs with indices five and higher do appear, and there are DAEs which have variable degeneracy, and they are non-reducible to the normal form.For singular differential equations with DI up to three effective specialized step-type methods exist [8]. In view of the p...

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The parameterization method in optimal control problems and differential–algebraic equations

Cited by 11 publications

References 4 publications

The Differential Transform Method for the Numerical Treatment of Differential-Algebraic Equations with Index 2

The Differential Transform Method for the Numerical Treatment of Differential-Algebraic Equations with Index 2

Dynamic sensitivity analysis of biological systems

Variational splines for the numerical solution of singular differential equations

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