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Erschienen in:
2013 | OriginalPaper | Buchkapitel
22. Necessary conditions
verfasst von : Francis Clarke
Erschienen in: Functional Analysis, Calculus of Variations and Optimal Control
Verlag: Springer London
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Abstract
Systems of ordinary differential equations such as x ′(t) = f(t,x(t)) are routinely used today to model a wide range of phenomena, in areas as diverse as aeronautics, power generation, robotics, economic growth, and natural resources. The great success of this paradigm is due in part to the fact that it suggests a natural mechanism through which the behavior of the system can be influenced by external factors. This is done by introducing an explicit control variable in the differential equation, a time-varying parameter that can be chosen (within certain limits) so as to attain a certain goal. This leads to the controlled differential equation
We present in this chapter the celebrated Pontryagin maximum principle, a set of necessary conditions for the optimal control of such a system.
$$ x\,' (t) \: = \:f( t, x(t) , u(t)) ,\,\:\: u(t)\in\, U. $$