标题： 最优控制和涉及$ψ$- Hilfer分数阶导数的分数阶半线性微分包含的近似能控性 显示英文标题

标题： Optimal Control and Approximate controllability of fractional semilinear differential inclusion involving $ψ$- Hilfer fractional derivatives

摘要： 本文首先研究了具有状态依赖控制约束的线性$\psi$-Hilfer分数阶导数的最优控制以及特定类型成本泛函的最优控制。然后，我们研究了在自反Banach空间中涉及$\psi$-Hilfer分数阶导数的抽象分数阶半线性微分包含的近似能控性。已知对于具有不同分数阶导数算子的类似类型的分数阶微分方程或包含，其存在性、唯一性、最优控制和近似能控性已被证明。因此，有必要研究包含所有具体分数阶导数算子的更一般的分数阶微分方程。这促使我们考虑$\psi$-Hilfer分数阶微分包含。我们假设相应半群的紧性和相关线性控制系统近似能控性，并借助对偶映射定义控制。我们观察到凸性在确定半线性微分包含的能控性性质中是必不可少的。在Hilbert空间的情况下，由于对偶映射变为简单的恒等映射，不存在凸性问题。与Hilbert空间相反，如果我们考虑自反Banach空间，由于对偶映射的非线性性质，会出现凸性问题。本文的新颖之处在于我们克服了这一凸性问题并建立了主要结果。最后，我们通过一个例子测试了我们的结果。 显示英文摘要

摘要： The current paper initially studies the optimal control of linear $\psi$-Hilfer fractional derivatives with state-dependent control constraints and optimal control for a particular type of cost functional. Then, we investigate the approximate controllability of the abstract fractional semilinear differential inclusion involving $\psi$-Hilfer fractional derivative in reflexive Banach spaces. It is known that the existence, uniqueness, optimal control, and approximate controllability of fractional differential equations or inclusions have been demonstrated for a similar type of fractional differential equations or inclusions with different fractional order derivative operators. Hence it has to research fractional differential equations with more general fractional operators which incorporate all the specific fractional derivative operators. This motivates us to consider the $\psi$-Hilfer fractional differential inclusion. We assume the compactness of the corresponding semigroup and the approximate controllability of the associated linear control system and define the control with the help of duality mapping. We observe that convexity is essential in determining the controllability property of semilinear differential inclusion. In the case of Hilbert spaces, there is no issue of convexity as the duality map becomes simply the identity map. In contrast to Hilbert spaces, if we consider reflexive Banach spaces, there is an issue of convexity due to the nonlinear nature of duality mapping. The novelty of this paper is that we overcome this convexity issue and establish our main result. Finally, we test our outcomes through an example.