View source for Variational problem

A variational problem with fixed ends is a problem in [[Variational calculus|variational calculus]] in which the end points of the curve which gives the extremum are fixed. Thus, in the simplest problem in variational calculus, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096260/v0962601.png" /> with fixed ends, the initial and final points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096260/v0962602.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096260/v0962603.png" /> through which the sought curve <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096260/v0962604.png" /> should pass are given. Since the general solution of the [[Euler equation|Euler equation]] of the simplest problem depends on two arbitrary constants, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096260/v0962605.png" />, the curve giving the extremum will be found among the solutions of the corresponding boundary value problem. It may turn out that the boundary value problem has only one solution, more than one solution or no solution at all.

A variational problem with free (mobile) ends is a problem in variational calculus in which the end points of the curve which gives the extremum may move along given manifolds. For instance, if in the [[Bolza problem|Bolza problem]] the number of boundary conditions to be satisfied by the sought curve <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096260/v0962606.png" /> is strictly less than <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096260/v0962607.png" />:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096260/v0962608.png" /></td> <td valign="top" style="width:5%;text-align:right;">(*)</td></tr></table>

the end points of the curve may move along the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096260/v0962609.png" />-dimensional manifold (*). If the boundary conditions (*) are given in the form

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096260/v09626010.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096260/v09626011.png" /></td> </tr></table>

and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096260/v09626012.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096260/v09626013.png" />, the end points of the curve <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096260/v09626014.png" /> may move along the respective manifolds of dimensions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096260/v09626015.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096260/v09626016.png" />. At the end points of the extremal curve the [[Transversality condition|transversality condition]] must be met; this, together with the conditions (*), makes it possible to obtain a closed system of relations leading to some boundary value problem. The solution of this boundary value problem yields arbitrary constants, which appear in the general integral of the Euler equation.

The qualitative difference between variational problems and the problem of finding extrema of a function of several variables consists in the fact that in the former case one is looking not for a point in a finite-dimensional space, but for a function (or a point in an infinite-dimensional space).



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====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  A.R.M. Noton,   "Introduction to variational methods in control engineering" , Pergamon  (1965)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  W.H. Fleming,   R.W. Rishel,   "Deterministic and stochastic optimal control" , Springer  (1975)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  L. Cesari,   "Optimization - Theory and applications" , Springer  (1983)</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  N.I. Akhiezer,   "The calculus of variations" , Blaisdell  (1962)  (Translated from Russian)</TD></TR></table>