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Difference methods for differential inclusions: A survey. (English) Zbl 0757.34018
In this survey the following initial value problem for ordinary differential inclusions is considered: “Let $$I=[t_ 0,T]$$ be a finite interval, $$y_ 0\in\mathbb{R}^ n$$, and $$F$$ be a map from $$I\times\mathbb{R}^ n$$ into the set of all subsets of $$\mathbb{R}^ n$$. Find an absolutely continuous function $$y(\cdot)$$ on $$I$$ such that $$y(t_ 0)=y_ 0$$ and $$\dot y(t)\in F(t,y(t))$$ for almost all $$t\in I$$, where $$\dot y(\cdot)$$ is the derivative of $$y(\cdot)$$.” Using difference method there exist various closely connected approaches of approximating solutions $$y(\cdot)$$. Investigations of convergence properties are presented, and applications to an example with discontinuous right-hand side are given. The classical Euler method is treated as an introductory example. Under the assumption of right-hand sides satisfying a one-sided Lipschitz condition Runge-Kutta schemes can be adapted to differential inclusions, too. The question, of whether the limit function $$y(\cdot)$$ has additional desirable properties, leads to selection strategies, which are illustrated by an example of a control system. Finally, error estimates and convergence properties of reachable sets are discussed. Many references are cited.

MSC:
 34A60 Ordinary differential inclusions 49M25 Discrete approximations in optimal control 65L05 Numerical methods for initial value problems involving ordinary differential equations 34-02 Research exposition (monographs, survey articles) pertaining to ordinary differential equations 65-02 Research exposition (monographs, survey articles) pertaining to numerical analysis 65J99 Numerical analysis in abstract spaces
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