Calculus of variations and optimal control; optimization Basic notions of the calculus of variations Concept
Functional
Summary
A functional is a map that takes an element of a vector space, usually a function, and returns a number. Length of a curve, energy of a field and the value of an integral are functionals of the curve, the field or the integrand. Functionals are the objects that the calculus of variations minimises and maximises. The linear ones, the linear forms, are the elements of the dual space and are central in functional analysis.
Where this leads
16 pages build on this one.
- Mayer and Lagrange problems
- Weak and strong extremum
- First variation
- Weak lower semicontinuity
- Topology optimization
- Bolza problem
- Integral functional with a Lagrangian density
- Hamilton's principle
- Coercive functional
- Stationary-action principle
- Shape derivative
- Second variation
- Relaxation of a variational problem
- Relaxed control
- Shape optimization
- Filippov's existence theorem
Further reading
About this page
| Tag | 1EQM0DJ4NC |
|---|---|
| URI | m2://1EQM0DJ4NC |
| Kind | concept |
| Writing | W0 stub |
| MSC | 49-XX, 46-XX |
| Same as |
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Used by
- Mayer and Lagrange problems
- Weak and strong extremum
- First variation
- Weak lower semicontinuity
- Bolza problem
- Integral functional with a Lagrangian density
- Coercive functional
- Stationary-action principle
- Shape derivative
- Second variation
- Relaxation of a variational problem
- Shape optimization