抽象分数阶时滞发展方程非局部问题可控性的研究

众所周知，分数阶时滞发展方程是指既含有关于时间变量t的分数阶导数，又含有时滞项的发展型偏微分方程。它既适合刻画多种材料或过程的记忆和遗传特征，又充分考虑了历史状态对当前状态的影响，因此研究分数阶时滞发展方程具有重要的理论意义和应用价值。本项目主要研究三类抽象分数阶时滞发展方程非局部问题的可控性：首先，研究非稠定情形下抽象分数阶时滞发展方程非局部问题的可控性，并将所获得的抽象结果应用于抛物型偏微分方程上，证明相应的可控性定理；其次，研究含扇形算子的分数阶时滞发展方程非局部问题的可控性，并将所获得的抽象结果应用于抛物型偏微分方程和反应扩散方程上，证明相应的可控性定理；最后，研究脉冲分数阶时滞发展方程非局部问题的可控性，并将所获得的抽象结果应用于含脉冲的抛物型偏微分方程和反应扩散方程上，证明相应的可控性定理。

It is well-known that the fractional order delay evolution equation is an evolution-type partial differential equation which contains both fractional order derivatives of time variables and delays. It is not only suitable for characterizing the memory and genetic characteristics of various materials or processes, but also fully considering the influence of historical state on the current state. Therefore, it is of great theoretical and practical value to study the fractional order delay evolution equation. In this project, we mainly study the controllability of nonlocal problems for three classes of abstract fractional order delay evolution equations. Firstly, we study the controllability of nonlocal problems of abstract fractional order delay evolution equations with nondensing domain. The obtained abstract results are applied to the parabolic-type partial differential equation to prove the corresponding controllability theorems. Secondly, we study the controllability of nonlocal problems of abstract fractional order delay evolution equations involving sectorial operator and apply the obtained abstract results to parabolic-type partial differential equation and reaction-diffusion equation to prove the corresponding controllability theorems. Finally, the controllability of nonlocal problems of abstract fractional evolution equations with impulses and delays is studied, and the obtained abstract results are applied to the parabolic-type partial differential equation and reaction-diffusion equation to prove the corresponding controllability theorems.