Calculus of variations and optimal control; optimization

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• ConceptCalculus of variations

  The calculus of variations studies the extreme values of functionals, that is, of maps that send functions to numbers. Examples of functionals are length, area and energy. A typical functional is an integral $\int_\Omega L(x, u, \nabla u) \, dx$, minimised among functions $u$ with given boundary values. Its smooth critical points solve a partial differential equation, the Euler–Lagrange equation. Conversely, many equations of physics and geometry are Euler–Lagrange equations, and the calculus of variations gives existence through the direct method.

• ConceptFunctional

  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.

• ConceptIntegral functional with a Lagrangian density

  An integral functional assigns to a function $u$ the integral of an expression $L(x, u, \nabla u)$ built from the point, the value of $u$ and its derivatives. The function $L$ is the Lagrangian density. Most problems of the calculus of variations concern such functionals, for example length, area, energy and action. Their smooth critical points solve the Euler–Lagrange equation.

• ConceptAdmissible function

  An admissible function of a variational problem is a function that is allowed to compete for the minimum. It must lie in the domain of the functional and satisfy every constraint of the problem, such as boundary values or a fixed length. The set of admissible functions fixes the problem as much as the functional does: a change of the class can change whether a minimiser exists.

• ConceptWeak and strong extremum

  In the calculus of variations a local minimum depends on what "nearby" means for functions. A strong local minimum beats every admissible function whose values are close. A weak local minimum only beats those whose values and slopes are both close. Every strong local extremum is a weak one, but not conversely. The Euler–Lagrange equation and the Legendre condition are necessary for weak extrema, while the Weierstrass excess function gives a further necessary condition for strong ones.

• ConceptFirst variation

  The first variation of a functional $J$ at $y$ in the direction $\eta$ is the rate of change of $J(y + t\eta)$ at $t = 0$. It is the analogue for functionals of the directional derivative. At a minimiser it vanishes in every admissible direction, and for integral functionals this condition leads to the Euler–Lagrange equation.