OPTIMALITY CONDITIONS OF FRACTIONAL DIFFUSION EQUATIONS WITH WEAK CAPUTO DERIVATIVES AND VARIATIONAL FORMULATION
OPTIMALITY CONDITIONS OF FRACTIONAL DIFFUSION EQUATIONS WITH WEAK CAPUTO DERIVATIVES AND VARIATIONAL FORMULATION
复制标题
弱Caputo导数分数扩散方程的最优条件及变分公式
DOI：
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发表时间：
2017
期刊：
影响因子：
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通讯作者：
Q. Tang
中科院分区：
文献类型：
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作者：
G. M. Bahaa;Q. Tang
在本文中，我们首先使用分布意义上的弱 Caputo 导数的新定义，并给出具有 Caputo 导数的分数扩散方程的变分公式。我们首先证明这个弱公式解的存在性，并用它来获得分布式边界分数最优控制问题（FOCP）的结果。然后我们证明所考虑的最优控制问题具有唯一解。 (FOCP) 的性能指数被视为状态变量和控制变量的函数，动态约束由偏分式微分方程 (PFDE) 表示。时间范围是固定的。我们对边界控制施加一些限制。用右分式弱Caputo导数定义的伴随问题解释Euler-Lagrange一阶最优性条件，得到最优控制的最优性系统。最后讨论了具有弱Caputo分数阶导数的分数分布Dirichlet问题的可控性。并结合一些例子进行了详细分析。
In this paper we start by using a new definition of weak Caputo derivative in the sense of distributions, and we give a variational formulation to a fractional diffusion equation with Caputo derivative. We first prove the existence of the solution to this weak formulation and use it to obtain a result on distributed and boundary Fractional Optimal Control Problem (FOCP). Then we show that the considered optimal control problem has a unique solution. The performance index of a (FOCP) is considered as a function of both state and control variables, and the dynamic constraints are expressed by a Partial Fractional Differential Equation (PFDE). The time horizon is fixed. We impose some constraints on the boundary control. Interpreting the Euler-?Lagrange first order optimality condition with an adjoint problem defined by means of right fractional weak Caputo derivative, we obtain an optimality system for the optimal control. Finally we discuss the controllability of the fractional distributed Dirichlet problem with weak Caputo fractional derivatives. Some examples are analyzed in details.