## 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.

### MSC:

 34A09 Implicit ordinary differential equations, differential-algebraic equations 49J05 Existence theories for free problems in one independent variable

### Keywords:

variational methods; convexity; coercivity; value function
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### References:

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