主题：Operator-valued, backward stochastic Riccati equations and application算子值倒向随机Riccati方程及其应用

主讲人：四川大学张旭教授

时间：2021年5月12日（星期三）16:30

地点：西南财经大学柳林校区通博楼B412

主办单位：经济数学学院 科研处

主讲人简介：

张旭，四川大学数学学院教授、博士生导师，主要研究数学控制论。曾独立获国家自然科学二等奖，获国家重点基础研究发展计划（973计划）、国家自然科学基金重点项目和国家杰出青年科学基金等资助，入选国家高层次人才特殊支持计划领军人才、四川省“天府万人计划”天府杰出科学家、教育部“创新团队发展计划”和中国科学院“百人计划”，入选“十一五”期间《国家自然科学基金资助项目优秀成果选编》和《国家杰出青年科学基金二十周年巡礼》。先后担任7种国际学术期刊的编委、副主编或主编，并应邀在国际数学家大会上作45分钟报告。

内容提要：

Riccati (type) equations date back to the very early period of modern mathematics. Some particular cases were studied more than three hundred years ago by J. Bernoulli and J. Riccati. Other important contributors on Riccati equations include D. Bernoulli, L. Euler, A.-M. Legendre, J. d’Alembert and so on. In the early stage, Riccati equations were in a narrow sense, i.e., first-order ordinary differential equations with quadratic unknowns. Later on, the term Riccati equation is also used to refer to matrix or operator equations with analogous quadratic unknowns. These equations appear in many different branches in mathematics. In particular, after R. E. Kalman's seminal work, the matrix-valued Riccati equations were extensively applied to solve many control problems. In this work, we introduce an operator-valued, backward stochastic Riccati equation for a general stochastic linear quadratic control problem in infinite dimensions. Generally speaking, the well-posedness of this equation is a challenging problem, even for a suitable notion of solutions to this equation. Indeed, in the infinite dimensional setting, there exists no satisfactory stochastic integration/evolution equation theory (in the literatures) which can be employed to treat the well-posedness of such a quadratically nonlinear equation. To overcome this difficulty, we adapt our transposition solution method, which was developed in our previous works but for operator-valued, backward stochastic (linear) Lyapunov equations. Riccati（型）方程可以追溯到现代数学的早期。我们对无限维一般随机线性二次控制问题，引入算子值倒向随机Riccati方程。一般说来，该方程的适定性极具挑战性。事实上，在无限维情形，还没有令人满意的随机积分/发展方程理论可以用来处理这类二次非线性方程的适定性。我们通过修改原本用于算子值倒向随机（线性）Lyapunov方程的转置解方法，来克服上述困难。