View source for Transversality condition

A necessary condition for optimality in variational problems with variable end-points. The arbitrary constants on which the solution of the [[Euler equation|Euler equation]] depends are determined by means of the tranversality condition. The transversality condition is a necessary condition for the vanishing of the first variation of a functional.

For the simplest problem in variational calculus with variable end-points,

<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/t/t094/t094010/t0940101.png" /></td> </tr></table>

in which the point

<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/t/t094/t094010/t0940102.png" /></td> </tr></table>

is not fixed but can belong to a certain manifold, the transversality condition can be written in the form of the relation

<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/t/t094/t094010/t0940103.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>

which must be satisfied for any values of the tangent differentials <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094010/t0940104.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094010/t0940105.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094010/t0940106.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094010/t0940107.png" /> of the boundary condition manifold.

If the left- and right-hand end-points of the extremal can be displaced along prescribed curves <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094010/t0940108.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094010/t0940109.png" />, then since

<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/t/t094/t094010/t09401010.png" /></td> </tr></table>

and the variations of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094010/t09401011.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094010/t09401012.png" /> are independent, (1) implies

<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/t/t094/t094010/t09401013.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2)</td></tr></table>

If the equations of the curves along which the left- and right-hand end-points are displaced are given in implicit form, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094010/t09401014.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094010/t09401015.png" />, then the transversality condition (1) can be written 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/t/t094/t094010/t09401016.png" /></td> <td valign="top" style="width:5%;text-align:right;">(3)</td></tr></table>

If there are no constraints on one of the end-points, then at this end-point, by virtue of the independence of the respective tangent differentials <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094010/t09401017.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094010/t09401018.png" />, the transversality condition takes 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/t/t094/t094010/t09401019.png" /></td> <td valign="top" style="width:5%;text-align:right;">(4)</td></tr></table>

The relations (2), (3), (4) are called transversality conditions.

Below, the transversality condition is given in the more general case of the variational problem for a conditional extremum. Consider the [[Bolza problem|Bolza problem]], i.e. the problem of minimizing a functional

<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/t/t094/t094010/t09401020.png" /></td> <td valign="top" style="width:5%;text-align:right;">(5)</td></tr></table>

in the presence of differential constraints of equality type,

<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/t/t094/t094010/t09401021.png" /></td> <td valign="top" style="width:5%;text-align:right;">(6)</td></tr></table>

and boundary conditions

<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/t/t094/t094010/t09401022.png" /></td> <td valign="top" style="width:5%;text-align:right;">(7)</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/t/t094/t094010/t09401023.png" /></td> </tr></table>

When <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094010/t09401024.png" /> in this problem, the end-points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094010/t09401025.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094010/t09401026.png" /> of the extremal are not fixed, but can be displaced along given hypersurfaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094010/t09401027.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094010/t09401028.png" />.

In accordance with the transversality condition, there exist constants (Lagrange multipliers) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094010/t09401029.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094010/t09401030.png" />, as well as multipliers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094010/t09401031.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094010/t09401032.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094010/t09401033.png" />, such that, in addition to the boundary conditions (7), the following relation holds at the end-points of the extremal:

<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/t/t094/t094010/t09401034.png" /></td> <td valign="top" style="width:5%;text-align:right;">(8)</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/t/t094/t094010/t09401035.png" /></td> </tr></table>

for any choice of tangent differentials

<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/t/t094/t094010/t09401036.png" /></td> <td valign="top" style="width:5%;text-align:right;">(9)</td></tr></table>

of the manifold defined by (7). In (8), <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094010/t09401037.png" /> denotes the expression

<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/t/t094/t094010/t09401038.png" /></td> <td valign="top" style="width:5%;text-align:right;">(10)</td></tr></table>

In the majority of practical problems, the Lagrange multipliers are normalized by setting <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094010/t09401039.png" /> (the value <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094010/t09401040.png" /> corresponds to an abnormal case, see [[#References|[1]]]). The multipliers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094010/t09401041.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094010/t09401042.png" />, are determined together with the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094010/t09401043.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094010/t09401044.png" />, from the solution of the Euler system of differential equations

<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/t/t094/t094010/t09401045.png" /></td> <td valign="top" style="width:5%;text-align:right;">(11)</td></tr></table>

and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094010/t09401046.png" /> equations of the form (6):

<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/t/t094/t094010/t09401047.png" /></td> </tr></table>

The general solution of this system of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094010/t09401048.png" /> second-order differential equations and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094010/t09401049.png" /> first-order differential equations in the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094010/t09401050.png" /> unknown functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094010/t09401051.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094010/t09401052.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094010/t09401053.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094010/t09401054.png" />, depends on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094010/t09401055.png" /> arbitrary constants. In fact, if one sets

<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/t/t094/t094010/t09401056.png" /></td> <td valign="top" style="width:5%;text-align:right;">(12)</td></tr></table>

then one obtains a system (11), (12) of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094010/t09401057.png" /> first-order differential equations and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094010/t09401058.png" /> finite relations

<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/t/t094/t094010/t09401059.png" /></td> <td valign="top" style="width:5%;text-align:right;">(13)</td></tr></table>

Using (13), some <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094010/t09401060.png" /> of the functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094010/t09401061.png" /> can be expressed in terms of the others (under the hypothesis that the corresponding functional determinant does not vanish) and, on substituting these in (11), (12), one obtains a system of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094010/t09401062.png" /> first-order differential equations with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094010/t09401063.png" /> unknown functions, the general solution of which depends on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094010/t09401064.png" /> arbitrary constants. Along with the values <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094010/t09401065.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094010/t09401066.png" />, this gives <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094010/t09401067.png" /> arbitrary constants, determining the solution of the variational problem (5)–(7). One then obtains by means of the transversality condition the correct number of equations enabling one to determine these arbitrary constants.

In problems of optimal control and in the [[Pontryagin maximum principle|Pontryagin maximum principle]], the necessary transversality condition is written similarly to (8), only instead of

<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/t/t094/t094010/t09401068.png" /></td> </tr></table>

one has to substitute in (8) the Hamiltonian <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094010/t09401069.png" />, taken with the opposite sign, and the conjugate variables <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094010/t09401070.png" />.

The necessary transversality condition gives the missing boundary conditions for obtaining a closed boundary value problem to which the solution of the variational problem with variable end-points reduces.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  G.A. Bliss,   "Lectures on the calculus of variations" , Chicago Univ. Press  (1947)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  M.A. Lavrent'ev,   L.A. Lyusternik,   "A course in variational calculus" , Moscow-Leningrad  (1950)  (In Russian)</TD></TR>
<TR><TD valign="top">[a1]</TD> <TD valign="top">  L. Cesari,   "Optimization - Theory and applications" , Springer  (1983)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  L.D. Berkovitz,   "Optimal control theory" , Springer  (1974)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  L.E. [L.E. El'sgol'ts] Elsgolc,   "Calculus of variations" , Pergamon  (1961)  (Translated from Russian)</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  R.H. Rishel,   "Deterministic and stochastic optimal control" , Springer  (1975)</TD></TR></table>

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