View source for Optimal trajectory

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

<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>

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>

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>

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

<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>

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>

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.

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" />:

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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" />.

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>

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>