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[Texts]## 49: Calculus of variations and optimal control; optimization |

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.

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.

- 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

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

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

GAMS listing of Optimal control software

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

- 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"?

Last modified 2000/01/14 by Dave Rusin. Mail: