Banach空间中时变随机广义发展方程的精确能控性问题研究

Since the publication of the paper "Control of randomly varying linear dynamic systems" by Kalman in 1962, the controllability of stochastic systems have become the central issues in the study of mathematical control theory, and a large number of academic papers have been published. Chinese scholars academician Chen Hanfu and academician Peng Shige have made important achievements in the research of these issues in the world. However, even for stochastic linear systems, there are still many important problems to be solved in terms of controllability. In particular, the exact controllability of general stochastic linear systems, which is very concerned by scholars in this field, is still unresolved. This project organizes a research group with outstanding young and middle-aged mathematical workers as the backbone, aiming at this mainstream scientific problem in mathematical control theory, jointly tackling key problems and focusing on regular discussion activities; Using infinite dimensional stochastic analysis as a tool, the exact controllability and observability of general time-varying stochastic singular evolution equations in Banach spaces are studied by using the theory of GE-evolution operators; Through the form of seminars, we gathered the strengths of all the experts. The sufficient and necessary conditions for exact controllability and observability of general time-varying stochastic singular evolution equations are obtained.

至从1962年Kalman发表论文“Control of randomly varying linear dynamical systems”以来，随机系统的能控性问题成为数学控制理论研究的中心问题，发表了大量学术论文。我国学者陈翰馥院士和彭实戈院士在这些问题的研究中取得了在国际上具有重要影响的研究成果。尽管如此，即使对于随机线性系统，就能控性而言，还有很多重要问题需要进一步解决。特别是国际上这一领域学者们十分关心的一般随机线性系统的精确能控性问题仍然悬而未决。本项目组织以优秀中青年数学工作者为骨干的研究小组，瞄准数学控制理论中的这一主流科学问题，联合攻关，集中开展定期研讨活动；以无限维随机分析为工具，利用广义发展算子理论研究Banach空间中一般时变随机广义发展方程的精确能控性和精确能观性；通过研讨班的形式集众家之长，获得一般时变随机广义发展方程精确能控和精确能观的充要条件。

至从1962年Kalman发表论文“Control of randomly varying linear dynamical systems”以来，随机系统的能控性问题成为数学控制理论研究的中心问题，发表了大量学术论文。我国学者陈翰馥院士和彭实戈院士在这些问题的研究中取得了在国际上具有重要影响的研究成果。尽管如此，即使对于随机线性系统，就能控性而言，还有很多重要问题需要进一步解决。特别是国际上这一领域学者们十分关心的一般随机线性系统的精确能控性问题仍然悬而未决。本项目组织以优秀中青年数学工作者为骨干的研究小组，瞄准数学控制理论中的这一主流科学问题，联合攻关，集中开展定期研讨活动；以无限维随机分析为工具，利用广义发展算子理论研究Banach空间中一般时变随机广义发展方程的精确能控性和精确能观性；通过研讨班的形式集众家之长。我们已获得了一般线性时变随机广义发展方程精确能控和精确能观的充要条件；半线性随机广义发展方程近似能控的充分条件；一般线性时变随机广义发展方程线性二次型最优控制问题解的充分条件。所得结果极大地丰富了随机广义系统的研究内容，对数学控制理论的发展有重要的推动作用。