A variational approach to implicit ODEs and differential inclusions. (English) Zbl 1172.34002

The authors study implicit differential equations of the form
\[ F(t,y(t),y'(t))=0. \]
They give sufficient conditions for \(F\) which ensure that global solutions exists. Additional conditions are provided which also ensures uniqueness of solutions. The new approach of this paper is to consider the differential equation as an optimization or variational problem, where the \(L^p\)-norm of \(F\) as a function of \(y\) is minimized. This approach is illustrated for the case of explicit differential equations and also for differential inclusions. It is suggested that this approach can be used to develop new methods for numerically approximating solutions of implicit differential equations. The paper is nicely written and well structured, however it should be noted that the so called coercivity condition is not fulfilled for the important class of linear implicit differential equations \(Ex'=Ax+f\) with singular matrix \(E\), so that the proposed framework can not be used for this case.


34A09 Implicit ordinary differential equations, differential-algebraic equations
49J05 Existence theories for free problems in one independent variable
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[1] P. Bochev and M. Gunzburger, Least-squares finite element methods. Proc. ICM2006III (2006) 1137-1162. · Zbl 1100.65098
[2] E.A. Coddington and N. Levinson, Theory of Ordinary Differential Equations. McGraw-Hill, New York (1955). · Zbl 0064.33002
[3] B. Dacorogna, Direct Methods in the Calculus of Variations. Springer (1989). · Zbl 0703.49001
[4] W.H. Fleming and H.M. Soner, Controlled Markov Processes and Viscosity Solutions. Springer, New York (2006). · Zbl 1105.60005
[5] J.D. Lambert, Numerical Methods for Ordinary Differential Systems: The Initial Value Problem. John Wiley and Sons Ltd. (1991). · Zbl 0745.65049
[6] G.V. Smirnov, Introduction to the Theory of Differential Inclusions, Graduate Studies in Mathematics41. American Mathematical Society (2002). · Zbl 0992.34001
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