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 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,

$$ J(x) = \int_{t_1}^{t_2} F(t, x, \dot x) \, dt, $$

in which the point

$$ (t_1, x(t_1), t_2, x(t_2)) = (t_1, x_1, t_2, x_2) $$

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

$$ [(F - \dot x F_{\dot x}) \ dt + F_{\dot x} \ dx]_1^2 = 0, $$

(1)

which must be satisfied for any values of the tangent differentials $d t_1$, $d x_1$, $dt_2$, $d x_2$ of the boundary condition manifold.

If the left- and right-hand end-points of the extremal can be displaced along prescribed curves $x = \phi_1(t)$ and $x = \phi_2(t)$, then since

$$d x_1 = \dot \phi_1(t) \, dt_1, \qquad d x_2 = \dot \phi_2(t) \, dt_2$$

and the variations of $dt_1$ and $dt_2$ are independent, (1) implies

$$ \left. \begin{array}{l} F(t_1, x_1, \dot x_1) - [\dot \phi_1(t_1) - \dot x_1] F_{\dot x} (t_1, x_1, \dot x_1) = 0, \\ F(t_2, x_2, \dot x_2) - [\dot \phi_2(t_2) - \dot x_2] F_{\dot x} (t_2, x_2, \dot x_2) = 0. \end{array} \right\} $$

(2)

If the equations of the curves along which the left- and right-hand end-points are displaced are given in implicit form, $\omega_1(t,x) = 0$ and $\omega_2(t,x) = 0$, then the transversality condition (1) can be written in the form

$$ \left. \begin{array}{ll} \dfrac{F - \dot x F_{\dot x}}{w_{1t}} = \dfrac{F{\dot x}}{\omega_{1x}} & \text{at the left-hand end-point,} \\ \dfrac{F - \dot x F_{\dot x}}{w_{2t}} = \dfrac{F{\dot x}}{\omega_{2x}} & \text{at the right-hand end-point} \end{array} \right\} $$

(3)

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 $dt$ and $dx$, the transversality condition takes the form

$$ F = 0, \qquad F_{\dot x} = 0. $$

(4)

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, i.e. the problem of minimizing a functional

$$ \begin{gathered} J(x) = \int_{t_1}^{t_2} f(t,x,\dot x) dt + g(t_1, x(t_1), t_2, x(t_2)), \\ f:\R \times \R^n \times \R^n \to \R, \qquad g:\R \times \R^n \times \R \times \R^n \to \R, \end{gathered} $$

(5)

in the presence of differential constraints of equality type,

$$ \phi(t,x,\dot x) = 0, \qquad \phi: \R\times \R^n\times \R^n\to\R^m, \quad m<n, $$

(6)

and boundary conditions

$$ \begin{gathered} \psi(t_1, x(t_1), t_2, x(t_2)) = 0,\\ \psi: \R \times \R^n\times\R\times\R^n\to\R^p, \qquad p\le 2n+2. \end{gathered} $$

(7)

When $p < 2n+2$ in this problem, the end-points $(t_1, x_1^1, \ldots, x_1^n)$ and $(t_2, x_2^1, \ldots, x_2^n)$ of the extremal are not fixed, but can be displaced along given hypersurfaces $\psi_\mu = 0$, $\mu = 1, \ldots, p$.

In accordance with the transversality condition, there exist constants (Lagrange multipliers) $e_\mu$, $\mu=1,\ldots,p$, as well as multipliers $\lambda_0$ and $\lambda_i(t)$, $i=1,\ldots,m$, such that, in addition to the boundary conditions (7), the following relation holds at the end-points of the extremal:

$$ \left[ \left( F - \sum_{i=1}^n \dot x^i F_{\dot x^i}\right) \ dt + \sum_{i=1}^n F_{\dot x^i} \ dx^i \right]_1^2 + \lambda_0 \ dg + \sum_{\mu=1}^p e_\mu \ d\psi_\mu =0, $$

(8)

for any choice of tangent differentials

$$ dt_1, dx_1^i, dt_2, dx_2^i, \qquad i = 1,\ldots, n, $$

(9)

of the manifold defined by (7). In (8), $F$ denotes the expression

$$ F = F(t,x,\dot x, \lambda) = \lambda_0 f(t,x,\dot x) + \sum_{i=1}^m \lambda_i(t) \phi_i(t, x, \dot x). $$

(10)

In the majority of practical problems, the Lagrange multipliers are normalized by setting $\lambda_0=1$ (the value $\lambda_0=0$ corresponds to an abnormal case, see [1]). The multipliers $\lambda_i(t)$, $i=1,\ldots,m$, are determined together with the $x^i(t)$, $i=1,\ldots,m$, from the solution of the Euler system of differential equations

$$ F_{x^i} - \frac{d}{dt} F_{\dot x^i} = 0, \qquad i = 1,\ldots,n, $$

(11)

and $m$ equations of the form (6):

$$ \phi_i(t,x,\dot x) = 0, \qquad i=1,\dots, m. $$

The general solution of this system of $n$ second-order differential equations and $m$ first-order differential equations in the $n+m$ unknown functions $x^i(t)$, $i=1,\ldots,n$, and $\lambda_i(t)$, $i=1,\ldots,m$, depends on $2n$ arbitrary constants. In fact, if one sets

$$ \dot x^i = u_i, \qquad i = 1,\ldots,n, $$

(12)

then one obtains a system (11), (12) of $2n$ first-order differential equations and $m$ finite relations

$$ \phi_i(t,x,u) = 0, \qquad i = 1,\ldots,m. $$

(13)

Using (13), some $m$ of the functions $u_i$ 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 $2n$ first-order differential equations with $2n$ unknown functions, the general solution of which depends on $2n$ arbitrary constants. Along with the values $t_1$ and $t_2$, this gives $2n+2$ 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, the necessary transversality condition is written similarly to (8), only instead of

$$ F - \sum_{i=1}^n \dot x^i F_{\dot x^i} \qquad \text{and} \qquad F_{\dot x^i} $$

one has to substitute in (8) the Hamiltonian $H$, taken with the opposite sign, and the conjugate variables $\psi_i$.

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

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