49: Calculus of variations and optimal control; optimization

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	ABOUT: [Introduction][History][Related areas][Subfields] POINTERS: [Texts][Software][Web links][Selected topics here] 49: Calculus of variations and optimal control; optimization

Introduction

Calculus of variations and optimization seek functions or geometric objects which are optimize some objective function. Certainly this includes a discussion of techniques to find the optima, such as successive approximations or linear programming. In addition, there is quite a lot of work establishing the existence of optima and characterizing them. In many cases, optimal functions or curves can be expressed as solutions to differential equations. Common applications include seeking curves and surfaces which are minimal in some sense. However, the spaces on which the analysis are done may represent configurations of some physical system, say, so that this field also applies to optimization problems in economics or control theory for example.

History

Applications and related fields

See also 65K (for numerical optimization), 90C (for applications of optimization to operations research, mathematical programming, etc.), 93 (for control theory in general), and 34H05 for differential equations in control problems. [Schematic of subareas and related areas]

Subfields

49J: Existence theory for optimal solutions
49K: Necessary conditions and sufficient conditions for optimality
49L: Carathéodory, Hamilton-Jacobi theories, including dynamic programming
49M: Methods of successive approximations, For discrete problems, see 90CXX; See also 65KXX
49N: Miscellaneous topics
49Q: Manifolds, see also 58FXX
49R50: Variational methods, For eigenvalues of operators, see 47A75
49S05: Variational principles of physics

The classification 49 was substantially re-organized in 1990, although almost all the changes were in the organization of the secondary levels; the extents of those secondary levels were not much changed. Roughly speaking, the old section 49A should cover the same terrain as 49J, and likewise we expect 49B=49K, 49C=49L, 49D=49M, 49E=49N, 49F=49Q, 49G=49R, and 49H=49S. Observe that the corresponding pairs are usually close in the map but not even contiguous in general. This heading thus provides a measure of the integrity of the mapping algorithms.

Browse all (old) classifications for this area at the AMS.

Textbooks, reference works, and tutorials

Pesch, Hans Josef: "A practical guide to the solution of real-life optimal control problems" Parametric optimization. Control Cybernet. 23 (1994), no. 1-2, 7--60. MR95b:49047

Bonnans, J. Frédéric; Shapiro, Alexander: "Optimization problems with perturbations: a guided tour", SIAM Rev. 40 (1998), no. 2, 228--264 (electronic). original article.

Fraser-Andrews, G.: "Finding candidate singular optimal controls: a state of the art survey", J. Optim. Theory Appl. 60 (1989), no. 2, 173--190. MR90b:49004

Software and tables

GAMS listing of Optimal control software

Other web sites with this focus

Here are the AMS and Goettingen resource pages for area 49.

Selected topics at this site

Calculus of Variations -- generalities and applications to particle motion (to minimize action)
Using Calculus of Variations to show straight lines are geodesics
A calculus of variations problem: find the curve of minimal length which joins two points and includes an area of 1.
What closed curve in R^3 has the smallest convex hull?
Choosing the path which maximizes average value of a function
Find the function u which minimizes an integral depending on u''
Possible answers to, "what curve is the seam on a baseball"?

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