Gevrey class explained

In mathematics, the Gevrey classes on a domain

\Omega\subseteq \R ^{n} , introduced by Maurice Gevrey,^[1] are spaces of functions 'between' the space of analytic functions C ^{\omega(\Omega)} and the space of smooth (infinitely differentiable) functions C ^{infty(\Omega)} . In particular, for \sigma \ge 1 , the Gevrey class G ^{\sigma} (\Omega) , consists of those smooth functions g \in C ^{infty(\Omega)} such that for every compact subset K \Subset \Omega there exists a constant C , depending only on g, K , such that^[2] \sup _{x} |D ^{\alpha} g(x)| \le C ^{|\alpha|+1} |\alpha!| ^{\sigma} \forall \alpha \in \Z ^{n} _{\geq 0} Where D ^{\alpha} denotes the partial derivative of order \alpha (see multi-index notation).

When

\sigma = 1 , G ^{\sigma(\Omega)} coincides with the class of analytic functions C ^{\omega(\Omega)} , but for \sigma > 1 there are compactly supported functions in the class that are not identically zero (an impossibility in C ^{\omega} ). It is in this sense that they interpolate between C ^{\omega} and C ^{infty} . 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 G ^{2(\Omega)} .

Application

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

\Phi _{\omega,T} (t) = \begin{cases} 0 & t \leq 0 , \\ 1 & t \geq T, \\ (\int ^{t} _{0} \Omega _{\omega,T} (\tau) d\tau)/(\int ^{T} _{0} \Omega _{\omega,T} (\tau) d\tau) & t \in ( 0 , T) \end{cases}

with

\Omega _{\omega,T} (t) = \begin{cases} 0 & t \notin [ 0 ,T], \\ \exp\left( (-1)/(\left([ 1 - (t)/(T) ] ~ (t)/(T) \right) ^{\omega}) \right) & t \in ( 0 , T) \end{cases}

and Gevrey order

\alpha = 1 + (1)/(\omega) .

See also

• Denjoy–Carleman theorem

Notes and References

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