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A direct method for the numerical solution of problems in the calculus of variations (cf. [[Variational calculus|Variational calculus]]). It is used for solving problems of [[Optimal control|optimal control]] of small dimensions, but with constraints on the phase coordinates and control functions. After discretization of the functional and the system of differential equations, the original problem is reduced to the minimization of a functional:

$$ \tag{1 }
I  =  \sum _ { k= } 0 ^ { N- }  1
F ^ { 0 } ( x _ {k} , x _ {k+} 1 , u _ {k} , t _ {k} , t _ {k+} 1 ),
$$

$$ \tag{2 }
x _ {k+} 1 - x _ {k}  =  \tau F( x _ {k} , x _ {k+} 1
, u _ {k} , t _ {k} , t _ {k+} 1 ),
$$

$$ 
k  =  0 \dots N - 1,
$$

$$ \tag{3 }
( x _ {k} , u _ {k} )  \in  G _ {k} ,\  k = 0 \dots N .
$$

Here  $  x _ {k} $, 
$  u _ {k} $
are vectors of the phase coordinates and the controls, respectively, at the node  $  t _ {k} $(
with respective dimensions  $  n $
and  $  m $), 
$  u _ {k} $
being considered as constant on each interval  $  ( t _ {k} , t _ {k+ 1 }  ) $, 
$  G _ {k} $
are given domains of an  $  ( n + m) $-
dimensional space ( $  G _ {0} $
and  $  G _ {N} $
describe the boundary conditions), and  $  \tau = ( T - t _ {0} )/N $
is the subdivision step of the initial interval  $  T - t _ {0} $.

The travelling-wave method is used in the case  $  n \geq  m $, 
which is typical of practical problems, and for which the use of other methods based on the variation of spatial states (cf. [[Travelling-tube method|Travelling-tube method]]; [[Local variations, method of|Local variations, method of]]) is difficult because of the labour involved in determining the control function.

The given initial approximation  $  ( x _ {0}  ^ {o} \dots x _ {N}  ^ {o} , u _ {0}  ^ {o} \dots u _ {N- 1 }   ^ {o} ) $, 
which satisfies (2) and (3), is improved in the sense of criterion (1) on each segment between  $  t _ {k} $
and  $  t _ {k+ p }  $(
$  x _ {k} $, 
$  x _ {k+ p }  $
are given), and this segment is gradually shifted by one node each time from the beginning of the trajectory to its end and back again (hence the name  "travelling wave" ).

One obtains, for each wave, a non-linear programming problem: The minimization of

$$ \tag{4 }
\Delta I _ {k}  = \ 
\sum _ { l= } k ^ { k+ }  p- 1
F ^ { 0 } ( x _ {l} , x _ {l+} 1 , u _ {l} , t _ {l} , t _ {l+} 1 )
$$

with  $  p $
constraints of the type (2) and under the conditions (3). In the practical realization of the travelling-wave method no attempt is made to solve problem (4); instead, the increments  $  \pm  h _ {i} $
are given to each one of the  $  r $
free parameters and, if  $  \Delta I _ {k} $(
in equation (4)) decreases and the conditions (3) are fulfilled, a new trajectory is obtained. If the trajectory remains unchanged during a complete period of the wave,  $  h _ {i} $
is replaced by a fraction of it.

If  $  m = n $, 
the travelling-wave method becomes identical with the method of local variations.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  N.N. Moiseev,   "Elements of the theory of optimal systems" , Moscow  (1975)  (In Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  I.A. Vatel,   A.F. Kononenko,   "A numerical scheme for solving optimal control problems"  ''USSR Comput. Math. Math. Phys.'' , '''10''' :  1  (1970)  pp. 85–94  ''Zh. Vychisl. Mat. i Mat. Fiz.'' , '''10''' :  1  (1970)  pp. 67–73</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  I.A. Vatel',   A.F. Kononenko,   ''Algoritm. i Program. (Informatsion. Byull.)'' :  2  (1972)  pp. 7</TD></TR></table>

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