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A curve  $  x( t) $
in an  $  ( n+ 1) $-
dimensional space of variables  $  t, x  ^ {1} \dots x  ^ {n} $
along which a point  $  x( t) = ( x  ^ {1} ( t) \dots x  ^ {n} ( t)) $, 
whose motion is determined by the vector differential equation

$$ \tag{1 }
\dot{x}  =  f( t, x, u),\  f:  \mathbf R \times \mathbf R  ^ {n} \times
\mathbf R  ^ {p} \rightarrow \mathbf R  ^ {n} ,
$$

is transferred from its original position

$$ \tag{2 }
x( t _ {0} )  =  x _ {0}  $$

to a final position

$$ \tag{3 }
x( t _ {1} )  =  x _ {1}  $$

under the influence of an [[Optimal control|optimal control]]  $  u( t) $
which minimizes a given functional

$$ \tag{4 }
J  =  \int\limits _ { t _ {0} } ^ { {t _ 1 } } f ^ { 0 }
( t, x, u)  dt,\ \ 
f ^ { 0 } :  \mathbf R \times \mathbf R  ^ {n} \times \mathbf R  ^ {p} \rightarrow
\mathbf R .
$$

The choice of an optimal control is subject to the restriction

$$ \tag{5 }
u  \in  U,
$$

where  $  U $
is a closed set of permissible controls,  $  U \subset  \mathbf R  ^ {p} $. 
The initial and final moments of time  $  t _ {0} $
and  $  t _ {1} $
are assumed to be fixed and free, respectively.

An optimal trajectory is defined in the same way for variational problems of a more general type than (1)–(5), for example, for problems with movable end-points and with constraints on the phase coordinates. For methods of tracing optimal trajectories, see [[Variational calculus, numerical methods of|Variational calculus, numerical methods of]].

For autonomous problems, in which the functions  $  f ^ { 0 } $, 
$  f $
do not explicitly depend on the time  $  t $:

$$ 
f ^ { 0 }  =  f  ^ {0} ( x, u),\ \ 
f  =  f( x, u),
$$

the concept of a phase optimal trajectory proves to be more apt for the theory and its applications. A phase optimal trajectory is the projection of an optimal trajectory onto the  $  n $-
dimensional subspace of phase variables  $  x  ^ {1} \dots x  ^ {n} $. 
For autonomous problems, a phase trajectory does not depend on the choice of the initial moment of time  $  t _ {0} $.

Research into the set of phase optimal trajectories which transfer the system from an arbitrary initial position to a given final position (or from a given initial position to an arbitrary final one) enables one to answer many qualitative questions arising from the variational problem being considered. The formation of the set of phase optimal trajectories is a compulsory step in the construction of a synthesis of an optimal feedback control

$$ 
u( t)  =  v( x( t)),
$$

which ensures a movement along an optimal trajectory at any point in the phase space.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  L.S. Pontryagin,   V.G. Boltayanskii,   R.V. Gamkrelidze,   E.F. Mishchenko,   "The mathematical theory of optimal processes" , Wiley  (1962)  (Translated from Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  P. DeRusso,   R. Roy,   C. Clois,   "State space in control theory" , Moscow  (1970)  (In Russian; translated from English)</TD></TR></table>

====Comments====

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  L. Markus,   "Foundations of optimal control theory" , Wiley  (1967)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  L. Cesari,   "Optimization - Theory and applications" , Springer  (1983)</TD></TR></table>