Lagrange problem

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

$$ J ( y) = \int\limits _ { x _ {1} } ^ { {x _ 2 } } f ( x , y , y ^ \prime ) d x ,\ \ f : \mathbf R \times \mathbf R ^ {n} \times \mathbf R ^ {n} \rightarrow \mathbf R , $$

in the presence of differential constraints of equality type:

$$ \tag{1 } \phi ( x , y , y ^ \prime ) = 0 ,\ \ \phi : \mathbf R \times \mathbf R ^ {n} \times \mathbf R ^ {n} \rightarrow \ \mathbf R ^ {m} ,\ m < n , $$

and boundary conditions

$$ \psi ( x _ {1} , y ( x _ {1} ) , x _ {2} , y ( x _ {2} ) ) = 0 ,\ \ \psi : \mathbf R \times \mathbf R ^ {n} \times \mathbf R \times \mathbf R ^ {n} \rightarrow \mathbf R ^ {p} , $$

$$ p \leq 2 n + 2 . $$

The Lagrange problem is usually considered under the condition that the system (1) is regular, that is, the matrix $ \| \partial \phi / \partial y ^ \prime \| $ has maximal rank:

$$ \mathop{\rm rank} \left \| \frac{\partial \phi }{\partial y ^ \prime } \right \| = m . $$

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

$$ \tag{2 } \left . \begin{array}{c} \int\limits _ { t _ {0} } ^ { {t _ 1 } } F ( t , x , u ) \ dt ,\ F : \mathbf R \times \mathbf R ^ {n} \times \mathbf R ^ {r} \rightarrow \mathbf R , \\ \dot{x} = \Phi ( t , x , u ) ,\ \Phi : \mathbf R \times \mathbf R ^ {n} \times \mathbf R ^ {r} \rightarrow \mathbf R ^ {n} . \\ \end{array} \right \} $$

The function $ F $ and the mapping $ \Phi $ 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 $ u \in U $. Necessary conditions for a strong extremum for the problem (2) (for simplicity, with fixed left-hand end $ x _ {0} $ and free right-hand end $ x _ {1} $) have the following form. Let

$$ L ( t , x , \dot{x} , u , p ( t) ) = \ ( p ( t) \mid - \dot{x} + \Phi ( t , x , u )) - F ( t , x , u ) $$

be the Lagrange function. For a vector function $ ( x ^ {*} ( t) , u ^ {*} ( t) ) $ to be a strong minimum in the Lagrange problem (2) it is necessary that the following relations hold:

$$ \tag{3 } \left . \frac{\partial L }{\partial \dot{x} } \right | _ {( x ^ {*} , u ^ {*} ) } + \int\limits _ { t _ {0} } ^ { {t _ 1 } } \left . \frac{\partial L }{d x } \right | _ {( x ^ {*} , u ^ {*} ) } d t = 0 , $$

$$ \tag{4 } p ( t _ {1} ) = 0 , $$

$$ \tag{5 } {\mathcal E} \equiv L ( t , x ^ {*} ( t) , \dot{x} , u , p ( t) ) + $$

$$ - L ( t , x ^ {*} ( t) , \dot{x} ^ {*} ( t) , u ^ {*} ( t) , p ( t) ) + $$

$$ - ( ( \dot{x} - \dot{x} ^ {*} ( t) ) \mid L _ {\dot{x} } ( t , x ^ {*} ( t ) , \dot{x} ^ {*} ( t ) , u ^ {*} ( t ) , p ( t )) ) = $$

$$ = \ ( p ( t) \mid \Phi ( t , x ^ {*} ( t) , u ) - F ( t , x ^ {*} ( t) , u )) + $$

$$ - ( p ( t) \mid \Phi ( t , x ^ {*} ( t) , u ^ {*} ( t) ) + F ( t , x ^ {*} ( t) , u ^ {*} ( t) )) \leq 0 $$

for all possible admissible values of $ \dot{x} $ and $ u $.

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

$$ {\mathcal H} ( t , x , u , p ) = ( p \mid \Phi ) - F , $$

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 $ ( x ^ {*} , u ^ {*} ) $ 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

$$ \dot{p} ( t) = - \frac{\partial {\mathcal H} ( t , x ^ {*} , u ^ {*} , p ) }{\partial x } ,\ p ( t _ {1} ) = 0 , $$

for which

$$ {\mathcal H} ( t , x ^ {*} ( t) , u ^ {*} ( t) , p ( t)) = \max _ {u \in U } {\mathcal H} ( t , x ^ {*} ( t) , u , p ( t) ) . $$

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 $ ( a \mid b) $ denotes the inner product of the vectors $ a $ and $ b $.