Gevrey class

In mathematics, the Gevrey classes on a domain $$\Omega\subseteq \R^n$$, introduced by Maurice Gevrey, 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
 * $$\sup_{x \in K} |D^\alpha g(x)| \le C^{|\alpha|+1}|\alpha!|^\sigma \quad \forall \alpha \in \Z_{\geq 0}^n$$

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. A typical example is the function


 * $$ \Phi_{\omega,T}(t) =

\begin{cases} 0 & t \leq 0, \\ 1 & t \geq T, \\ \frac{\int_{0}^{t} \Omega_{\omega,T}(\tau) d\tau}{\int_{0}^{T} \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( \frac{-1}{\left([1 - \frac{t}{T}] ~ \frac{t}{T} \right)^{\omega}} \right)  & t \in (0, T) \end{cases} $$

and Gevrey order $$ \alpha = 1 + \frac{1}{\omega}.$$