Lagrange problem

One of the fundamental problems in the classical calculus of variations. It consists in minimizing the functional

in the presence of differential constraints of equality type:

(1)

and boundary conditions

The Lagrange problem is usually considered under the condition that the system (1) is regular, that is, the matrix has maximal rank:

Under this condition the system (1) can be solved for part of the variables and, using a different notation ( instead of ), the Lagrange problem can be reduced to the form

(2)

The function and the mapping are usually assumed to be continuously differentiable. Problems of optimal control are often specified in the form (2) (the Pontryagin form), and restrictions are, moreover, imposed on the control . Necessary conditions for a strong extremum for the problem (2) (for simplicity, with fixed left-hand end and free right-hand end ) have the following form. Let

be the Lagrange function. For a vector function to be a strong minimum in the Lagrange problem (2) it is necessary that the following relations hold:

(3)

(4)

(5)

for all possible admissible values of and .

If one carries out differentiation in (3) with respect to and uses the notation

then a necessary condition for a strong minimum can be stated in the form of a maximum principle, in which the Euler equation (3), the transversality condition (4) and the Weierstrass condition (5) are combined. For a vector function to be a strong minimum in the problem (2) with fixed left-hand end and free right-hand end it is necessary that there is a solution of the system

for which

J.L. Lagrange considered similar problems in connection with studies in mechanics (in the second half of the 18th century).

For references see Variational calculus.

Comments

The notation denotes the inner product of the vectors and .