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GeezerVison is showing mathworld.wolfram.com — Bolza Problem -- from Wolfram MathWorld

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Bolza Problem

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 y(t) 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.
(1)

Here L is the Lagrangian function and Phi is the endpoint cost, also called the Mayer term. In the general formulation, y(t) can have several components, the endpoint times can vary, and admissible curves can obey differential equations phi(t,y,y^')=0, constraints psi(t,y)=0, and constraints chi(t_0,y(t_0),t_1,y(t_1))=0 on the endpoints. An admissible curve satisfies the stipulated regularity, differential equation, and endpoint conditions. The prime in y^'(t) denotes the derivative with respect to t. The Lagrange form has Phi=0, and the Mayer form has L=0 (Liberzon 2012, §3.3.2).

For the scalar problem with fixed times t_0 and t_1, free values y_0=y(t_0) and y_1=y(t_1), and no other constraints, suppose L and y are twice continuously differentiable and Phi is continuously differentiable. Stationarity gives the Euler-Lagrange differential equation in the interior. At the endpoints the first variation gives the natural boundary conditions

L_(y^')(t_0,y_0,y^'(t_0))=(partialPhi)/(partialy_0)
(2)
L_(y^')(t_1,y_1,y^'(t_1))=-(partialPhi)/(partialy_1).
(3)

The opposite signs come from evaluating the boundary term [L_(y^')h]_(t_0)^(t_1) after integration by parts, where h(t) is the function used to vary y(t) to y(t)+epsilonh(t). This boundary term comes from varying the integral, while Phi is a separate part of the original cost. When Phi=0, these reduce to L_(y^')=0 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.

See also

Calculus of Variations, Euler-Lagrange Differential Equation, Functional

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References

Bliss, G. A. "The Problem of Bolza in the Calculus of Variations." Ann. Math. 33, 261-274, 1932. https://doi.org/10.2307/1968328.Gelfand, I. M. and Fomin, S. V. Calculus of Variations. Englewood Cliffs, NJ: Prentice-Hall, 1963.Goldstine, H. H. A History of the Calculus of Variations from the 17th through the 19th Century. New York: Springer-Verlag, p. 374, 1980.Liberzon, D. "Cost Functional." §3.3.2 in Calculus of Variations and Optimal Control Theory: A Concise Introduction. Princeton, NJ: Princeton University Press, 2012. https://liberzon.csl.illinois.edu/teaching/cvoc/node52.html.

Referenced on Wolfram|Alpha

Bolza Problem

Cite this as:

Weisstein, Eric W. "Bolza Problem." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/BolzaProblem.html

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