## Optimal control problem for a degenerate fractional differential equation.(English)Zbl 1468.49020

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.

### MSC:

 49K15 Optimality conditions for problems involving ordinary differential equations 93C10 Nonlinear systems in control theory 93C15 Control/observation systems governed by ordinary differential equations
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