Conditions for Strong Minimum of Functional

Theorem
Let $ \mathbf y $ be an $ n $-dimensional vector such that $ \mathbf y \left ( { a } \right ) = A $ and $ \mathbf y \left ( { b } \right ) = B $

Let $ J $ be a functional such that:


 * $ \displaystyle J \left [ { \mathbf y } \right ] = \int_a^b F \left ( { x, \mathbf y, \mathbf y' } \right) \mathrm d x $

Let $ \gamma $ be an extremal curve of $ J $.

Let the following be the field of the functional $ J $:


 * $ \mathbf y' = \boldsymbol \psi \left ( { x, \mathbf y } \right ) $

Let $ R $ be an open region containing $ \gamma $ and have the field $ \boldsymbol \psi $ defined $ \forall \left ( { x, \mathbf y } \right ) \in R $.

Let $ \mathbf w $ be a finite vector.

Suppose that


 * $ \forall \left ( { x, \mathbf y } \right ) \in R : E \left ( { x, \mathbf y, \boldsymbol \psi, \mathbf w } \right ) \ge 0 $

where $ E $ is Weierstrass E-Function.

Then $ J $ has a strong minimum for $ \gamma $.

Proof
By definition, the increment of $ J $ is:


 * $ \displaystyle \Delta J = \int_{ \gamma ^* } F \left ( { x, \mathbf y, \mathbf y' } \right ) \mathrm d x - \int_{ \gamma } F \left ( { x, \mathbf y, \mathbf y' } \right ) \mathrm d x $

where $ \gamma $ and $ \gamma^* $ are curves described by $ \left ( { x, \mathbf y \left ( { x } \right ) } \right ) $ and $ \left ( { x, \mathbf y^* \left ( { x } \right ) } \right ) $ respectively, such that $ \mathbf y^* \left ( { x } \right ) - \mathbf y \left ( { x } \right ) = \mathbf h \left ( { x } \right ) $.

Consider Hilbert's invariant integral:

Since the integrand is full differential, the integral does not depend on the shape of $ \Gamma $, but only on its endpoints.

Therefore, for $ \Gamma = \gamma $ and $ \Gamma = \gamma^* $ the value of the integral is the same.

Since $ \mathbf y' \left ( { x } \right ) = \boldsymbol \psi \left ( { x, \mathbf y } \right ) $ determines boundary conditions for $ \gamma $, $ \Gamma = \gamma $ is one of the trajectories of the field $ \mathbf y' \left ( { x } \right ) = \boldsymbol \psi \left ( { x, \mathbf y } \right ) $.

Hence, $ \mathrm d \mathbf y $ is constrained by $ \mathrm d \mathbf y = \boldsymbol \psi \mathrm d x $:


 * $ \displaystyle g \left ( { x, \mathbf y } \right ) = \int_\gamma F \left ( { x, \mathbf y, \mathbf y' } \right ) \mathrm d x $

Thus, $ g \left ( { x, \mathbf y' } \right ) $ can be written in two different ways.


 * $ \displaystyle \int_\gamma F \left ( { x, \mathbf y, \mathbf y' } \right ) \mathrm d x = \int_{ \gamma^* } \left ( { \{ { F \left ( { x, \mathbf y, \boldsymbol \psi } \right ) - \boldsymbol \psi F_{ \mathbf y' } \left ( { x, \mathbf y, \boldsymbol \psi } \right ) } \} \mathrm d x + F_{ \mathbf y' } \left ( { x, \mathbf y, \boldsymbol \psi } \right ) \mathrm d \mathbf y } \right ) $

Substitute this into the expression for $ \Delta J $:

By assumption:


 * $ E \left ( { x, \mathbf y, \boldsymbol \psi, \mathbf y' } \right ) \ge 0 $

Hence the integrand is bounded below by $ 0 $ and above by its maximum value in the interval of integration.

Hence, the integral is bounded below by $ 0 $ and above by some positive number.

Therefore:


 * $ \Delta J \ge 0 $.