A variational difference scheme for a boundary value problem with a small parameter in the highest derivative

Boglaev, Igor

doi:10.1016/0041-5553(81)90036-7

Cited by 22 publications

(10 citation statements)

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“…The decomposition at the construction of the observer reduces actually to the decomposition of the asymptotic stability property in the system of the form (9) and to the possibility of the selection of the matrices L I and L a which ensure the decomposition of the asymptotic stability property.…”

Section: Controllability~ Observability and St A>ilizabilitymentioning

confidence: 99%

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Singular perturbations in optimal control problems

Vasil’eva¹,

Dmitriev²

1986

J Math Sci

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One presents a survey of the works on the theory of optimal control by deterministic objects, described by systems of ordinary differential or difference equations, where the investigation is carried out with the aid of the methods of the theory of singular perturbations. One analyzes the possibility of the application of the theory of singular perturbations to the investigation of control problems with large amplification coefficient in the feedback circuit, to the description of sliding regimes in systems with variable structure, and to the construction of effective numerical algorithms for the solution of optimization problems. INTRODUCTIONIn the last decades the mathematical theory of optimal processes has been intensively developed. Considerable attention has been given to problems with small perturbations. The application of the methods of perturbation theory (asymptotic methods) in problems of optimal control is useful because of the following reasons.i. In mathematical modeling one neglects frequently various small quantities and as a result one operates with idealized models~ The theory of perturbations allows us to establish a correspondence between the exact (perturbed) and the simplified (unperturbed, degenerate) models. This correspondence is established by the investigation of a limiting process from the solution of the perturbed problem to the solution of the unperturbed one. The theory of perturbations allows us to obtain corrections to the solution of the simplified problem (asymptotic expansion) and allows us to study the sensitivity of the solution of the exact problem with respect to small perturbations.2. The asymptotic theory gives us the possibility to obtain a qualitative picture of the solution~ 3. The knowledge of the as~cmptotics allows us to offer economical computational procedures for the approximate solution of the initial problems of optimal control, the solution of which is difficult or practically impossible because of nonlinearity, computational instability (as a consequence of the "rigidity"), high dimensionality, etco 4. Small perturbations in optimal control problems can be introduced artifically and then the theory of perturbations emerges as a method of investigation of the initial, in a certain sense "bad" (for example, ill-posed) problem.The investigation of optimal control problems by the methods of perturbation theory presents interest also for perturbation theory itself since its development is stimulated by the various new problems which emerge. The survey of F. L. Chernous'ko and V. B. Kolmanovskii The present survey is devoted to deterministic objects described by systems of ordinary differential or difference equations. Stochastic objects and objects described by partial differential equations are virtually not considered in this survey. The given bibliography does not have the pretention of absolute completeness. Additional references can be found in the works cited in the survey. We give some definitions and notations which will be used in the survey. The symbol ...

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Section: Controllability~ Observability and St A>ilizabilitymentioning

confidence: 99%

“…The proof of the fact that (II) is actually the problem P0 reduces to the proof of the fact that the boundary value problem of the maximum principle, written for (II), coincides with the degenerate problem (8), (9). In the case of a linear term outside the integral, there arise some particularities [62].…”

Section: O=nz=~i~ (Hzz--ii'zhy:h"z)mentioning

confidence: 99%

Singular perturbations in optimal control problems

Vasil’eva¹,

Dmitriev²

1986

J Math Sci

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“…Earlier these types of problems have been solved by numerous researchers. Boglave [1] and Schatz and Wahlbin [11] used finite element techniques to solve such problems. Niijima [7,8] gave uniformly second-order accurate difference schemes, whereas Miller [6] gave sufficient conditions for the uniform first-order convergence of a general three-point difference scheme.…”

Section: Introductionmentioning

confidence: 99%

Initial-value technique for self-adjoint singular perturbation boundary value problems

2009

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We have developed an initial-value technique for self-adjoint singularly perturbed two-point boundary value problems. The original problem is reduced to its normal form, and the reduced problem is converted into first-order initial-value problems. These initial-value problems are solved by the cubic spline method. Numerical illustrations are given at the end to demonstrate the efficiency of our method. Graphs are also depicted in support of the results.

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“…The crux of the proof lies in suitably bounding the discretized Green's function and its derivative. Using exponential trial functions to interpolate between these nodal values, our approximation is then shown to be uniformly first-order accurate in L°° [0,1]. Thus the second-order nodal accuracy is a superconvergence result.…”

mentioning

confidence: 99%

“…Hegarty et al [4] and Niijima [7] produced uniformly second-order difference schemes. Boglaev [1] examined problem (1.2) in a finite-element framework and achieved uniform first-order accuracy at the nodes. Shishkin [11] examined problem (1.1) on a nonuniform mesh, which depends on e, and obtained convergence results for various difference schemes.…”

mentioning

confidence: 99%

A uniformly accurate finite-element method for a singularly perturbed one-dimensional reaction-diffusion problem

O’Riordan¹,

Stynes²

1986

Math. Comp.

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Abstract. A finite-element method with exponential basis elements is applied to a selfadjoint, singularly perturbed, two-point boundary value problem. The tridiagonal difference scheme generated is shown to be uniformly second-order accurate for this problem (i.e., the nodal errors are bounded by Ch2, where C is independent of the mesh size h and the perturbation parameter). With a certain choice of trial functions, uniform first-order accuracy is obtained in /. K[0,l].

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A variational difference scheme for a boundary value problem with a small parameter in the highest derivative

Cited by 22 publications

References 1 publication

Singular perturbations in optimal control problems

Singular perturbations in optimal control problems

Initial-value technique for self-adjoint singular perturbation boundary value problems

A uniformly accurate finite-element method for a singularly perturbed one-dimensional reaction-diffusion problem

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