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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.

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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