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# 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.

## 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.

## 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
• 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