Discontinuous variational problem

A problem in the calculus of variations in which an extremum of a functional is attained on a polygonal extremal. A polygonal extremal is a piecewise-smooth solution of the Euler equation satisfying certain additional necessary conditions at the vertices. The actual form of these conditions depends on the type of the discontinuous variational problem. Thus, in a first-order discontinuous variational problem the polygonal extremal is found by making the usual assumptions of continuity and continuous differentiability of the integrand. For the simplest kind of functional

$$\tag{1 } J = \int\limits _ { x _ {1} } ^ { {x _ 2 } } F ( x, y, y ^ \prime ) dx,\ \ y ( x _ {1} ) = y _ {1} ,\ \ y ( x _ {2} ) = y _ {2} ,$$

it is necessary that the Weierstrass–Erdmann conditions

$$\tag{2 } F _ {y ^ \prime } ( x _ {0} , y ( x _ {0} ),\ y ^ \prime ( x _ {0} - 0)) = \ F _ {y ^ \prime } ( x _ {0} , y ( x _ {0} ),\ y ^ \prime ( x _ {0} + 0)),$$

$$\tag{3 } F ( x _ {0} , y ( x _ {0} ), y ^ \prime ( x _ {0} + 0)) -$$

$$- y ^ \prime ( x _ {0} - 0) F _ {y ^ \prime } ( x _ {0} , y ( x _ {0} ), y ^ \prime ( x _ {0} - 0)) =$$

$$= \ F ( x _ {0} , y ( x _ {0} ), y ^ \prime ( x _ {0} + 0)) + y ^ \prime ( x _ {0} + 0) F _ {y ^ \prime } ( x _ {0} , y ( x _ {0} ), y ^ \prime ( x _ {0} + 0))$$

be fulfilled at a corner $x _ {0}$ of the polygonal extremal. When $F$ depends on $n$ unknown functions, that is, when $y$ in (1) is an $n$- dimensional vector $y = ( y _ {1} \dots y _ {n} )$, then the Weierstrass–Erdmann corner conditions analogous to (2), (3) are

$$\tag{4 } \left [ \frac{\partial F }{\partial y _ {i} ^ \prime } \right ] _ {x _ {0} - 0 } = \ \left [ \frac{\partial F }{\partial y _ {i} ^ \prime } \right ] _ {x _ {0} + 0 } ,\ \ i = 1 \dots n,$$

$$\tag{5 } \left [ F - \sum _ {i = 1 } ^ { n } y _ {i} ^ \prime \frac{\partial F }{\partial y _ {i} ^ \prime } \right ] _ {x _ {0} - 0 } = \left [ F - \sum _ {i = 1 } ^ { n } y _ {i} ^ \prime \frac{\partial F }{\partial y _ {i} ^ \prime } \right ] _ {x _ {0} + 0 } .$$

For problems on conditional extrema in which the integrand depends on $n$ unknown functions and when there are $m$ differential constraints given as equations (see Bolza problem), the Weierstrass–Erdmann conditions have to be expressed in terms of the Lagrange function $L$ and have the same form as (4), (5), but with $F$ replaced by $L$.

In terms of the theory of optimal control the necessary conditions at a corner of a polygonal extremal require the continuity of the conjugate variables and of the Hamilton function at the points of discontinuity of the optimal control. As is implied by the Pontryagin maximum principle, these conditions are automatically fulfilled if along a polygonal extremal the control is determined by the condition that the Hamilton function has a maximum.

In a second-order discontinuous variational problem the integrand is discontinuous. Let, for example, $F ( x, y, y ^ \prime )$ have a discontinuity along the line $y = \phi ( x)$, so that $F ( x, y, y ^ \prime )$ is equal to $F _ {1} ( x, y, y ^ \prime )$ and $F _ {2} ( x, y, y ^ \prime )$, respectively, along one side or the other of $y = \phi ( x)$. Then, if the optimal solution exists, it is achieved on a polygonal extremal which has a corner at $( x _ {0} , \phi ( x _ {0} ))$ and one obtains, instead of the functional (1), the functional

$$\tag{6 } J = \int\limits _ { x _ {1} } ^ { {x _ 0 } } F _ {1} ( x, y, y ^ \prime ) dx + \int\limits _ { x _ {0} } ^ { {x _ 2 } } F _ {2} ( x, y, y ^ \prime ) dx = \ J _ {1} + J _ {2} .$$

A variation of the functional (6) reduces to a variation of the functionals $J _ {1}$ and $J _ {2}$ on matching curves which have moving right and left end points sliding along $y = \phi ( x)$. In order that a minimum for the functional (6) is attained on a polygonal extremal, it is necessary that at a corner $( x _ {0} , \phi ( x _ {0} ))$ one has

$$\tag{7 } [ F _ {1} - ( \phi ^ \prime - y ^ \prime ) F _ {1y ^ \prime } ] _ {x = x _ {0} - 0 } = \ [ F _ {2} + ( \phi ^ \prime - y ^ \prime ) F _ {2y ^ \prime } ] _ {x = x _ {0} + 0 } .$$

When $F$ depends on $n$ unknown functions $y = ( y _ {1} \dots y _ {n} )$ and the surface of discontinuity of $F$ is given in the form

$$\tag{8 } \Phi ( x, y) = 0,$$

the necessary conditions at a corner of a polygonal extremal which is on the surface (8) take the form

$$\tag{9 } \frac{\left [ F _ {1} - \sum _ {i = 1 } ^ { n } \frac{\partial F _ {1} }{\partial y _ {i} ^ \prime } \right ] _ {x _ {0} - 0 } - \left [ F _ {2} - \sum _ {i = 1 } ^ { n } \frac{\partial F _ {2} }{\partial y _ {i} ^ \prime } \right ] _ {x _ {0} + 0 } }{\left [ \frac{\partial \Phi }{\partial x } \right ] _ {x _ {0} } } =$$

$$= \ \frac{\left [ \frac{\partial F _ {1} }{\partial y _ {1} ^ \prime } \right ] _ {x _ {0} - 0 } - \left [ \frac{\partial F _ {2} }{\partial y _ {1} ^ \prime } \right ] _ {x _ {0} + 0 } }{\left [ \frac{\partial \Phi }{\partial y _ {1} } \right ] _ {x _ {0} } } = \dots$$

$$\dots = \frac{\left [ \frac{\partial F _ {1} }{\partial y _ {n} ^ \prime } \right ] _ {x _ {0} - 0 } - \left [ \frac{\partial F _ {2} }{\partial y _ {n} ^ \prime } \right ] _ {x _ {0} + 0 } }{\left [ \frac{\partial \Phi }{\partial y _ {n} } \right ] _ {x _ {0} } } .$$

The necessary conditions (7), (9) are insufficient for computing the arbitrary constants determining the polygonal extremal — it is a particular solution of the Euler equation satisfying the boundary conditions. In fact, the equations (9) give $n$ necessary conditions which, together with the $2n$ boundary conditions, the $n$ conditions for the polygonal extremal to be joined continuously at a corner and equation (8) give $4n + 1$ conditions, so that it is possible to determine the $x$- coordinate $x _ {0}$ of the vertex and $4n$ arbitrary constants, $2n$ for each of the extremals coming up to the different sides of the surface (8).

References

 [1] N.M. Gyunter, "A course in the calculus of variations" , Leningrad-Moscow (1941) (In Russian) [2] V.I. Smirnov, "A course of higher mathematics" , 4 , Addison-Wesley (1964) (Translated from Russian) [3] L.S. Pontryagin, V.G. Boltayanskii, R.V. Gamkrelidze, E.F. Mishchenko, "The mathematical theory of optimal processes" , Wiley (1962) (Translated from Russian)