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A curve <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068510/o0685101.png" /> in an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068510/o0685102.png" />-dimensional space of variables <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068510/o0685103.png" /> along which a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068510/o0685104.png" />, whose motion is determined by the vector differential equation
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<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/o/o068/o068510/o0685105.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>
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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
 
is transferred from its original position
  
<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/o/o068/o068510/o0685106.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2)</td></tr></table>
+
$$ \tag{2 }
 +
x( t _ {0} )  = x _ {0}  $$
  
 
to a final position
 
to a final position
  
<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/o/o068/o068510/o0685107.png" /></td> <td valign="top" style="width:5%;text-align:right;">(3)</td></tr></table>
+
$$ \tag{3 }
 +
x( t _ {1} ) =  x _ {1}  $$
  
under the influence of an [[Optimal control|optimal control]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068510/o0685108.png" /> which minimizes a given functional
+
under the influence of an [[Optimal control|optimal control]] $  u( t) $
 +
which minimizes a given 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/o/o068/o068510/o0685109.png" /></td> <td valign="top" style="width:5%;text-align:right;">(4)</td></tr></table>
+
$$ \tag{4 }
 +
= \int\limits _ { t _ {0} } ^ { {t _ 1 } } f ^ { 0 }
 +
( t, x, u)  dt,\ \
 +
f ^ { 0 } : \mathbf R \times \mathbf R  ^ {n} \times \mathbf R  ^ {p} \rightarrow
 +
\mathbf R .
 +
$$
  
 
The choice of an optimal control is subject to the restriction
 
The choice of an optimal control is subject to the restriction
  
<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/o/o068/o068510/o06851010.png" /></td> <td valign="top" style="width:5%;text-align:right;">(5)</td></tr></table>
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$$ \tag{5 }
 +
u  \in  U,
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068510/o06851011.png" /> is a closed set of permissible controls, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068510/o06851012.png" />. The initial and final moments of time <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068510/o06851013.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068510/o06851014.png" /> are assumed to be fixed and free, respectively.
+
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]].
 
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068510/o06851015.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068510/o06851016.png" /> do not explicitly depend on the time <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068510/o06851017.png" />:
+
For autonomous problems, in which the functions $  f ^ { 0 } $,  
 +
$  f $
 +
do not explicitly depend on the time $  t $:
  
<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/o/o068/o068510/o06851018.png" /></td> </tr></table>
+
$$
 +
f ^ { 0 }  = f  ^ {0} ( x, u),\ \
 +
= f( x, u),
 +
$$
  
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068510/o06851019.png" />-dimensional subspace of phase variables <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068510/o06851020.png" />. For autonomous problems, a phase trajectory does not depend on the choice of the initial moment of time <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068510/o06851021.png" />.
+
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
 
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
  
<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/o/o068/o068510/o06851022.png" /></td> </tr></table>
+
$$
 +
u( t)  = v( x( t)),
 +
$$
  
 
which ensures a movement along an optimal trajectory at any point in the phase space.
 
which ensures a movement along an optimal trajectory at any point in the phase space.
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====References====
 
====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>
 
<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====
 
====Comments====
 
  
 
====References====
 
====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>
 
<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>

Latest revision as of 08:04, 6 June 2020


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 $ 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.

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

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

Comments

References

[a1] L. Markus, "Foundations of optimal control theory" , Wiley (1967)
[a2] L. Cesari, "Optimization - Theory and applications" , Springer (1983)
How to Cite This Entry:
Optimal trajectory. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Optimal_trajectory&oldid=13714
This article was adapted from an original article by I.B. Vapnyarskii (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article