Difference methods for differential inclusions

F. Lempio: Difference methods for differential inclusions
in: Modern Methods of Optimization. Proceedings of a Summer School at Thurnau (Germany), September 30-October 6, 1990, Lecture Notes in Economics and Mathematical Systems 378, Springer Verlag, Berlin-Heidelberg-New York, 1992, 236 - 273

DOI: 10.1007/978-3-662-02851-3_8
Zentralblattnummer: 0762.65042
Keywords: difference methods; differential inclusions; convergence; linear multistep method; initial value problem; Euler method; numerical examples
Mathematics Subject Classification Code: 65L05 (65L06 65L12 34A34)


First we introduce differential inclusions by means of several model problems. These model problems shall illustrate the significance of differential inclusions for a wide range of applications, e.g. dynamic systems with discontinuous state equations, nonlinear programming, and optimal control.

Then we concentrate on difference methods for initial value problems. The basic convergence proof for linear multistep methods is given. Main emphasis is laid on the fundamental ideas behind the proof techniques in order to clarify the meaning of all relevant assumptions. Especially, instead of global boundedness of the right-hand side we prefer imposing a growth condition, and moreover examine the influence of errors thoroughly.

Finally, we outline order of convergence proofs for differential inclusions satisfying a one-sided Lipschitz condition. For higher dimensional problems the underlying difference methods must satisfy consistency and stability properties familiar from ordinary stiff differential equations and not shared by explicit methods. Nevertheless, we can clarify the proof structure already by the classical explicit Euler method for one-dimensional problems. Thus, by the way we prove first order convergence of Euler method for special problems not necessarily satisfying the Lipschitz condition


1. Introduction
2. Convergent Difference Approximations
3. Higher Order Convergence

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