一类二阶中立型随机积微分方程的精确可控性

Exact Controllability for a Class of Second-Order Neutral Stochastic Integrodifferential Equations

  • 摘要: 在研究积微分发展方程时,系统可能会因白噪声或其他随机因素的影响而产生扰动,这类随机因素会对模型的运行结果造成影响。同时,在某些时刻受到外界影响时,系统会发生瞬时状态变化,虽然这种变化相较于整个时间演化过程极为短暂,但这种瞬时变化可能会影响模型的整体表现和最终结果,这类现象即为脉冲现象。因此,建立具有瞬时脉冲的随机积微分方程模型是很有必要的。本文研究Hilbert空间中一类具有非局部条件和瞬时脉冲的二阶中立型随机积微分方程的精确可控性。由于该方程不含控制项,控制算子满足的约束条件单独给出,因而我们首先运用变量代换的技巧和强连续余弦算子族的相关理论,重新定义了新的算子,得到了所研究问题适度解的表达式,并为后续得到精确可控性建立了理论基础。随后,通过利用随机分析理论、H \ddot\mathrmo lder不等式和Sadovskii不动点定理等数学工具,得到了所研究问题适度解的存在性结果和精确可控性的充分条件。所获结论进一步丰富了积微分发展方程的理论研究。

     

    Abstract: In the study of integrodifferential evolution equations, systems may be disturbed by white noise or other stochastic factors, which affect the operational results of the model. Meanwhile, when subjected to external influences at certain moments, the system will undergo instantaneous state changes. Although such changes are extremely short-lived compared to the entire time evolution process, they may affect the overall performance and final results of the model, this phenomenon is known as the impulse phenomenon. Therefore, it is necessary to establish a stochastic integrodifferential equation model with instantaneous impulses. In this paper, we investigate the exact controllability of a class of second-order neutral stochastic integrodifferential equations with nonlocal conditions and instantaneous impulses in Hilbert spaces. Since the equation does not contain a control term, the constraint conditions satisfied by the control operator are given separately. Thus, we first use the technique of variable substitution and the relevant theory of strongly continuous cosine operator families to redefine new operators, get the express of the mild solution of the studied problem and also establishes a theoretical foundation for the subsequent exact controllability. Subsequently, the existence result of mild solutions and sufficient conditions for the exact controllability of the studied problem are derived by employing mathematical tools such as the theory of stochastic analysis, H \ddot\mathrmo lder inequality and Sadovskii fixed point theorem. The results obtained further enrich the theoretical research on integrodifferential evolution equations.

     

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