The Bolza problem is a problem in the calculus of variations that seeks a minimum of a functional
combining an integral cost along an admissible curve 
with a cost depending on its endpoints
(Bliss 1932, Goldstine 1980, p. 374),
![J[y]=Phi(t_0,y(t_0),t_1,y(t_1))+int_(t_0)^(t_1)L(t,y(t),y^'(t))dt.](/images/equations/BolzaProblem/NumberedEquation1.svg) |
(1)
|
Here 
is the Lagrangian function and 
is the endpoint cost, also called the Mayer
term. In the general formulation, 
can have several components, the endpoint times can vary,
and admissible curves can obey differential
equations 
, constraints 
, and constraints 
on the endpoints.
An admissible curve satisfies the stipulated regularity,
differential equation, and endpoint
conditions. The prime in 
denotes the derivative with respect to 
. The Lagrange form has 
, and the Mayer form
has 
(Liberzon 2012, §3.3.2).
For the scalar problem with fixed times 
and 
, free values 
and 
, and no other constraints,
suppose 
and 
are twice continuously differentiable
and 
is continuously differentiable.
Stationarity gives the Euler-Lagrange
differential equation in the interior. At the endpoints the first variation
gives the natural boundary conditions
 |  |  |
(2)
|
 |  |  |
(3)
|
The opposite signs come from evaluating the boundary term ![[L_(y^')h]_(t_0)^(t_1)](/images/equations/BolzaProblem/Inline25.svg)
after integration
by parts, where 
is the function used to vary 
to 
. This boundary
term comes from varying the integral, while 
is a separate part of the original cost. When 
, these reduce to 
at both endpoints (Gelfand and Fomin 1963, pp. 24-26).
The symbolic solver EulerLagrangeSolve in a future version of the Wolfram Language will accept a Bolza cost specified by its integrand and endpoint term.
