Gevrey class

In mathematics, the Gevrey classes on a domain $Ω \subseteq ℝ^{n}$ , introduced by Maurice Gevrey, are spaces of functions 'between' the space of analytic functions $C^{ω} (Ω)$ and the space of smooth (infinitely differentiable) functions $C^{\infty} (Ω)$ . In particular, for $σ \geq 1$ , the Gevrey class $G^{σ} (Ω)$ , consists of those smooth functions $g \in C^{\infty} (Ω)$ such that for every compact subset $K ⋐ Ω$ there exists a constant $C$ , depending only on $g, K$ , such that

sup_{x \in K} | D^{α} g (x) | \leq C^{| α | + 1} | α! |^{σ} \forall α \in ℤ_{\geq 0}^{n}

Where $D^{α}$ denotes the partial derivative of order $α$ (see multi-index notation).

When $σ = 1$ , $G^{σ} (Ω)$ coincides with the class of analytic functions $C^{ω} (Ω)$ , but for $σ > 1$ there are compactly supported functions in the class that are not identically zero (an impossibility in $C^{ω}$ ). It is in this sense that they interpolate between $C^{ω}$ 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 $G^{2} (Ω)$ .

Application

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

Φ_{ω, T} (t) = {\begin{matrix} 0 & t \leq 0, \\ 1 & t \geq T, \\ \frac{\int_{0}^{t} Ω_{ω, T} (τ) d τ}{\int_{0}^{T} Ω_{ω, T} (τ) d τ} & t \in (0, T) \end{matrix}

with

Ω_{ω, T} (t) = {\begin{matrix} 0 & t \notin [0, T], \\ \exp (\frac{- 1}{{([1 - \frac{t}{T}] \frac{t}{T})}^{ω}}) & t \in (0, T) \end{matrix}

and Gevrey order $α = 1 + \frac{1}{ω} .$

References

↑ 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.
1 2 Rodino, L. (Luigi) (1993). Linear partial differential operators in Gevrey spaces. Singapore: World Scientific. ISBN 981-02-0845-6. OCLC 28693208.
↑ Schaum, Alexander; Meurer, Thomas (2020). Control of PDE systems (lecture notes).
↑ 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.

[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.

[:0-2] 1 2 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.

Gevrey class

Application

See also

References