Optimal control problem for a degenerate fractional differential equation
The classical Riemann-Liouville left fractional derivative is \[ ^{RL} D^\alpha_{0+} f(t) = \frac{1}{\Gamma(n-\alpha)} \Big(\frac{d}{dt} \Big)^n \int_0^t (t - \tau)^{n - \alpha - 1} f(\tau) d\tau \qquad (n-1 < \alpha \le n) \, , \] its right counterpart based on the integration interval \(t \le \tau \le T.\) The Caputo left fractional derivative uses instead the definition \[ ^{C} D^\alpha_{0+} f(t) = \frac{1}{ \Gamma(n-\alpha)} \int_0^t (t - \tau)^{n - \alpha - 1} f^{(n)}(\tau) d\tau \qquad (n-1 < \alpha \le n) \, , \] its right counterpart based on \(t \le \tau \le T.\) Both the \(RL\) and the \(C\) derivatives reduce to the ordinary derivative when \(\alpha = n.\) One advantage of the second definion is, the \(C\) derivative of any order \(\alpha\) of a constant is zero, while we have \(^{RL} D^\alpha_{0-}c = (c t^{n-\alpha}/\Gamma(n - \alpha + 1))^{(n)},\) which is zero only for integer values of \(\alpha.\) Another is, the Laplace transform of Caputo \(\alpha^{th}\) derivatives involves \(f(0), \dots, f^{(n-1)}(0)\) rather than fractional derivatives of \(f(\cdot)\) at \(t = 0\), thus allowing pointwise initial conditions in fractional differential equations. The control systems in this paper are of the form \[ v(t) ^{C} D_{0+}^\alpha u(t) + a(t)u(t) = \varphi(t, \nu(t)) \, \quad (0 \le t \le T), \qquad u(0) = \varphi_0 \] where \(0 < \alpha \le 1,\) \(\nu(t) = (\nu_1(t), \dots \nu_m(t))\) is an \(m\)-dimensional control function in a control set \(\Omega\) and \(v(t)\) is positive and locally integrable. The problem is to minimize the cost functional \[ F(\nu) = \sum_{k=1}^N \alpha_k u(t^{(k)}) \] where the \(\{t^{(k)}\}\) are given in \([0, T].\) The authors derive the adjoint equation and obtain Pontryagin's maximum principle for the optimal control \(\bar \nu\) in the usual maximization-of-Hamiltonian form \[ H(t, f_\alpha(t), \bar \nu(t)) = \max_{\nu \in \Omega}H(t, f_\alpha(t), \nu) \] where \(f_\alpha(t)\) is the solution of the adjoint equation and \(H(t, f_\alpha(t), \nu) = f_\alpha (t) \cdot \varphi(t, \nu)\) is the Hamiltonian.
degenerate fractional optimal control problem
scientific article; zbMATH DE number 7369521
scientific article