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Erschienen in:
2013 | OriginalPaper | Buchkapitel
25. Differential inclusions
verfasst von : Francis Clarke
Erschienen in: Functional Analysis, Calculus of Variations and Optimal Control
Verlag: Springer London
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Abstract
We develop necessary conditions for the optimal control of systems governed by a differential inclusion, in the context of the following problem: This differential inclusion problem seems far less natural than the standard control formulation, and it does not lend itself very well to modeling. Nonetheless, it provides an ideal mathematical environment in which to prove various types of necessary conditions. All the different versions of the maximum principle presented so far have followed from the extended maximum principle; this, in turn, will be a consequence of the theorems proved in this chapter.
$$ \left\{ \begin{array}{ll} \quad \text{Minimize } \quad &J(x)\: =\: \displaystyle{\ell\big( x(a) , x(b)\big)} \\ \quad \text{subject to} &x \,' (t)\: \in \:F_{\,t}\big( x(t)\big) ,\;\; t\in\, [\,a ,b\,] {~\,\text {a.e.}}\\ &\big( x(a) , x(b)\big)\in\, E . \end{array} \right. $$