Filippov Solutions to Systems of Ordinary Differential Equations with Delta Function Terms as Summands

  • Uladzimir HrusheuskiEmail author
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 44)


This paper is devoted to the investigation of the Cauchy problem for the system of ordinary differential equations
$$ \dot{y}(t) = f \bigl(t,y(t)\bigr) + A{\delta}^{(s)}(t), \quad y(-1) = y_0 \in {\mathbb{R}}^n, $$
with a vector containing derivatives of the delta function and a possibly discontinuous function \(f:[-1,T_{0}] \times{\mathbb{R}}^{n} \rightarrow{\mathbb{R}}^{n}\), T 0>0, and a constant matrix A on the right-hand side. In our approach, the components of δ (s) are replaced by derivatives of different δ-sequences and the limiting behavior of approximating solutions is examined. Filippov’s notion of solution to a differential equation with discontinuous right-hand side is used.


Delta function Differential equations with distributions Differential inclusions Differential equations with discontinuous right-hand sides Filippov solution 

Mathematics Subject Classification

34A36 34A37 34A60 46T30 34A26 



The author expresses his heartfelt gratitude to all members of the Unit of Engineering Mathematics, University of Innsbruck, for their hospitality and friendly atmosphere and especially their leader, Michael Oberguggenberger, for his encouragement, fruitful discussions during the meetings and attentive reading of the manuscript. Special thanks are to Peter Wagner for all the time spent working together and Anna Bombasaro for her help in administrative questions. The author also thanks the Austrian Agency for International Cooperation in Education and Research (OeAD) and Austrian Federal Ministry of Science and Research (BMWF) for the financial support and help in organizational matters.


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© Springer International Publishing Switzerland 2013

Authors and Affiliations

  1. 1.Unit of Engineering MathematicsUniversity of InnsbruckInnsbruckAustria

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