This paper deals with fractional optimal control governed by semilinear equations using the increment approach. We have considered controlled object as

$$({\mathrm{\Gamma }}_{\alpha }s)(\tau )\equiv {}^C{D}_{0+}^{\alpha }s(\tau )-B(\tau )s(\tau )=\eta (\tau ,C(\tau )),\tau \in (0,T),$$

with the initial conditions:

$${\mathrm{\Gamma }}_{0}s\equiv s(0)=s_0,$$

$${\mathrm{\Gamma }}_{1}s\equiv s^{\prime }(0)=s_1,$$

where α ∈ (1, 2], s(τ) is state variable in $ℝ$, B(τ) ∈ L_p, ϱ(0, T), and $s_0,s_1\in ℝ$. Let the m-dimensional control vector function be C(τ) and defined as $C(\tau )=(C_1(\tau ),\dots ,C_m(\tau ))$. Assume the function η(τ, C(τ)) satisfies Caratheodory condition and defined on $(0,\tau )\times {ℝ}^m$. We have obtained our results with the help of the adjoint equation and Pontryagin's maximum condition. For better understanding, we have included one example.

中文翻译：

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分数阶 (1, 2] 非线性系统最优控制问题的一种新增量方法

本文使用增量方法处理由半线性方程控制的分数最优控制。我们将受控对象视为

$$({\mathrm{\Gamma }}_{\alpha }s)(\tau )\equiv {}^C{D}_{0+}^{\alpha }s(\tau )-乙(\tau )s(\tau )=\eta (\tau ,C(\tau )),\tau \in (0,吨),$$

初始条件：

$${\mathrm{\Gamma }}_{0}s\equiv s(0)=s_0,$$

$${\mathrm{\Gamma }}_{1}s\equiv s^{\text{'}}(0)=s_1,$$

其中α  ∈ (1, 2], s ( τ ) 是状态变量$ℝ$, B ( τ ) ∈  L _{p ,  ϱ} (0,  T ) 和$s_0,s_1\in ℝ$. 设m维控制向量函数为C ( τ ) 并定义为$C(\tau )=(C_1(\tau ),\dots ,C_{米}(\tau ))$. 假设函数η ( τ ,  C ( τ )) 满足 Caratheodory 条件并定义在$(0,\tau )\times {ℝ}^{米}$. 我们在伴随方程和 Pontryagin 最大条件的帮助下得到了我们的结果。为了更好地理解，我们举了一个例子。