Pointwise completeness and pointwise degeneracy of fractional standard and descriptor linear continuous-time systems with different fractional orders
The subject is continuous time linear systems for vector functions \(x(t)\) containing fractional derivatives \((d^\alpha /dt^\alpha) x(t)\). The derivative is the \textit{Caputo fractional derivative} \[ \frac{d^\alpha}{dt^\alpha} x(t) = \frac{1}{\Gamma(n - \alpha)} \int_0^t \frac{1}{(t - \tau)^{\alpha - n + 1}} \frac{d^n}{d\tau^n} x(\tau) d\tau \quad (n - 1 < \alpha < n) \, . \tag{1} \] The choice of this derivative over the Riemann-Liouville version \[ \frac{d^\alpha}{dt^\alpha} x(t) = \frac{1}{\Gamma(n - \alpha)} \frac{d^n}{dt^n}\int_0^t \frac{1}{(t - \tau)^{\alpha - n + 1}} x(\tau) d\tau \quad (n - 1 < \alpha < n) \tag{2} \] is motivated by the fact that the Laplace transform of (1) involves \(x(0), \dots, x^{(n-1)}(0)\) rather than fractional derivatives of \(x(\cdot)\) at \(t = 0\) as in (2), thus allowing pointwise initial conditions. The particular system examined is \[ E \left[ \begin{matrix} \frac{d^\alpha}{dt^\alpha} x_1(t) \\ \frac{d^\beta}{dt^\beta} x_2(t) \end{matrix} \right] = A \left[ \begin{matrix} x_1(t) \\ x_2(t) \end{matrix} \right] \tag{3} \] for \(x(t) = (x_1(t), x_2(t))\). The system (3) is \textit{pointwise complete} for \(t = t_f\) if for every \(x_f\) there exists initial conditions such that the solution \(x(\cdot)\) satisfies \(x(t_f) = x_f\). The results are on completeness and on its absence, degeneracy. Roughly, they are the counterpart for systems of those in [\textit{T. Kaczorek} and \textit{A. Ruszewski}, Int. J. Appl. Math. Comput. Sci. 30, No. 2, 219--223 (2020; Zbl 1465.93095)] for the one-derivative system \(E (d^\alpha / dt^\alpha )x(t) = Ax(t)\).
scientific article; zbMATH DE number 7342320
scientific article