Gevrey class
In mathematics, the Gevrey classes on a domain [math]\displaystyle{ \Omega\subseteq \R^n }[/math], introduced by Maurice Gevrey,[1] are spaces of functions 'between' the space of analytic functions [math]\displaystyle{ C^\omega(\Omega) }[/math] and the space of smooth (infinitely differentiable) functions [math]\displaystyle{ C^\infty(\Omega) }[/math]. In particular, for [math]\displaystyle{ \sigma \ge 1 }[/math], the Gevrey class [math]\displaystyle{ G^\sigma (\Omega) }[/math], consists of those smooth functions [math]\displaystyle{ g \in C^\infty(\Omega) }[/math] such that for every compact subset [math]\displaystyle{ K \Subset \Omega }[/math] there exists a constant [math]\displaystyle{ C }[/math], depending only on [math]\displaystyle{ g, K }[/math], such that[2]
- [math]\displaystyle{ \sup_{x \in K} |D^\alpha g(x)| \le C^{|\alpha|+1}|\alpha!|^\sigma \quad \forall \alpha \in \Z_{\geq 0}^n }[/math]
Where [math]\displaystyle{ D^\alpha }[/math] denotes the partial derivative of order [math]\displaystyle{ \alpha }[/math] (see multi-index notation).
When [math]\displaystyle{ \sigma = 1 }[/math], [math]\displaystyle{ G^\sigma(\Omega) }[/math] coincides with the class of analytic functions [math]\displaystyle{ C^\omega(\Omega) }[/math], but for [math]\displaystyle{ \sigma \gt 1 }[/math] there are compactly supported functions in the class that are not identically zero (an impossibility in [math]\displaystyle{ C^\omega }[/math]). It is in this sense that they interpolate between [math]\displaystyle{ C^\omega }[/math] and [math]\displaystyle{ C^\infty }[/math]. 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 [math]\displaystyle{ G^2(\Omega) }[/math].[2]
Application
Gevrey functions are used in control engineering for trajectory planning.[3] [4] A typical example is the function
- [math]\displaystyle{ \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} }[/math]
with
- [math]\displaystyle{ \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} }[/math]
and Gevrey order [math]\displaystyle{ \alpha = 1 + \frac{1}{\omega}. }[/math]
See also
- Denjoy–Carleman theorem
References
- ↑ Gevrey, Maurice (1918). "Sur la nature analytique des solutions des équations aux dérivées partielles. Premier mémoire" (in en). Annales scientifiques de l'École Normale Supérieure 35: 129–190. doi:10.24033/asens.706. http://www.numdam.org/item/?id=ASENS_1918_3_35__129_0.
- ↑ 2.0 2.1 Rodino, L. (Luigi) (1993). Linear partial differential operators in Gevrey spaces. Singapore: World Scientific. ISBN 981-02-0845-6. OCLC 28693208. https://www.worldcat.org/oclc/28693208.
- ↑ Schaum, Alexander; Meurer, Thomas (2020). Control of PDE systems (lecture notes). https://www.control.tf.uni-kiel.de/en/teaching/summer-term/control-of-pde-systems/fileadmin/lecture_notes_2020.
- ↑ 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) (Bologna (Italy)) 43 (14): 587–592. doi:10.3182/20100901-3-IT-2016.00215. https://www.sciencedirect.com/science/article/pii/S1474667015370270.
Original source: https://en.wikipedia.org/wiki/Gevrey class.
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