In mathematics, the Gevrey classes on a domain , introduced by Maurice Gevrey,[1] are spaces of functions 'between' the space of analytic functions and the space of smooth (infinitely differentiable) functions . In particular, for , the Gevrey class , consists of those smooth functions such that for every compact subset there exists a constant , depending only on , such that[2]

Where denotes the partial derivative of order (see multi-index notation).

When , coincides with the class of analytic functions , but for there are compactly supported functions in the class that are not identically zero (an impossibility in ). It is in this sense that they interpolate between and . The Gevrey classes find application in discussing the smoothness of solutions to certain partial differential equations: Gevrey originally formulated the definition while investigating the homogeneous heat equation, whose solutions are in .[2]

Application

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Gevrey functions are used in control engineering for trajectory planning.[3] [4] A typical example is the function

 

with

 

and Gevrey order  

See also

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References

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  1. ^ Gevrey, Maurice (1918). "Sur la nature analytique des solutions des équations aux dérivées partielles. Premier mémoire". Annales scientifiques de l'École Normale Supérieure. 35: 129–190. doi:10.24033/asens.706.
  2. ^ a b Rodino, L. (Luigi) (1993). Linear partial differential operators in Gevrey spaces. Singapore: World Scientific. ISBN 981-02-0845-6. OCLC 28693208.
  3. ^ Schaum, Alexander; Meurer, Thomas (2020). Control of PDE systems (lecture notes).
  4. ^ Utz, Tilman; Graichen, Knut; Kugi, Andreas (2010). "Trajectory planning and receding horizon tracking control of a quasilinear diffusion-convection-reaction system". Proceedings 8th IFAC Symposium "Nonlinear Control Systems" (NOLCOS). 43 (14). Bologna (Italy): 587–592. doi:10.3182/20100901-3-IT-2016.00215.