Continuation and path following

Allgower, Eugene L.; Georg, Kurt

doi:10.1017/s0962492900002336

Cited by 210 publications

(156 citation statements)

References 249 publications

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“…There exists a well-developed theory and many algorithms and numerical methods implementing these ideas, and the field of applications encompasses Brouwer fixed point problems, polynomial and nonlinear systems of equations, boundary value problems in many diverse forms, etc. We refer the reader to the survey articles [161,162,163,164] or to the textbook [165] for a complete report on these theories and methods.…”

Section: The Continuation Methodsmentioning

confidence: 99%

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Optimal Control and Applications to Aerospace: Some Results and Challenges

Trélat

2012

J Optim Theory Appl

187

179

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This article surveys the classical techniques of nonlinear optimal control such as the Pontryagin Maximum Principle and the conjugate point theory, and how they can be implemented numerically, with a special focus on applications to aerospace problems. In practice the knowledge resulting from the maximum principle is often insufficient for solving the problem in particular because of the well-known problem of initializing adequately the shooting method. In this survey article it is explained how the classical tools of optimal control can be combined with other mathematical techniques to improve significantly their performances and widen their domain of application. The focus is put onto three important issues. The first is geometric optimal control, which is a theory that has emerged in the 80's and is combining optimal control with various concepts of differential geometry, the ultimate objective being to derive optimal synthesis results for general classes of control systems. Its applicability and relevance is demonstrated on the problem of atmospheric re-entry of a space shuttle. The second is the powerful continuation or homotopy method, consisting of deforming continuously a problem towards a simpler one, and then of solving a series of parametrized problems to end up with the solution of the initial problem. After having recalled its mathematical foundations, it is shown how to combine successfully this method with the shooting method on several aerospace problems such as the orbit transfer problem. The third one consists of concepts of dynamical system theory, providing evidence of nice properties of the celestial dynamics that are of great interest for future mission design such as low cost interplanetary space missions. The article ends with open problems and perspectives.

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Section: The Continuation Methodsmentioning

confidence: 99%

“…It can be noted that the feasibility of the continuation method has been much studied for other less specific issues in numerical analysis (see e.g. [170,171,161,162,163] and references therein).…”

mentioning

confidence: 99%

Optimal Control and Applications to Aerospace: Some Results and Challenges

Trélat

2012

J Optim Theory Appl

187

179

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“…The method inherits the flexibility of numerical continuation methods (see [1,11]), while remedying a number of weaknesses such as danger of breakdown at steep turns of ∂Λ (A) and lack of large grain parallelism.…”

Section: Domain-based Methodsmentioning

confidence: 99%

The design of a distributed MATLAB-based environment for computing pseudospectra

Bekas

Kokiopoulou

Gallopoulos

2005

Future Generation Computer Systems

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“…Such methods are becoming a viable alternative in circuit simulation, where they can be used to resolve convergence difficulties and to find multiple operation points. These methods are slower than conventional methods but their speed can often be improved by careful implementation [12][13][14][15][16][17].…”

Section: Introductionmentioning

confidence: 99%

Spherical Continuation Algorithm with Spheres of Variable Radius to Trace Homotopy Curves

Torres-Muñoz

Hernandez-Martinez

Vázquez-Leal

2015

Int. J. Appl. Comput. Math

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The homotopy continuation methods are capable to locate multiple solutions of nonlinear systems of equations. In this paper is presented the path-following technique used to trace the homotopy trajectory. The spherical method is employed resizing the radius of the sphere at each iteration, the above based on the behavior of the radius of curvature. Also the Newton homotopy is applied in conjunction with the proposed methodology for tracing homotopy curve. To prove the usefulness of the proposed method is applied to three examples with different numbers of equations obtaining successful results.

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Continuation and path following

Cited by 210 publications

References 249 publications

Optimal Control and Applications to Aerospace: Some Results and Challenges

Optimal Control and Applications to Aerospace: Some Results and Challenges

The design of a distributed MATLAB-based environment for computing pseudospectra

Spherical Continuation Algorithm with Spheres of Variable Radius to Trace Homotopy Curves

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