发表机构
Fudan University; Shenzhen University（复旦大学; 深圳大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
研究次无限时域随机线性二次最优控制问题，通过引入新型带延迟项的黎卡提方程，克服求解困难，找到时间一致等均衡策略，建立比较定理并给出方程解在\(T\to\infty\)时的收敛行为。
AI 中文摘要
本文研究一类次无限时域随机线性二次最优控制问题，初始时间\(t\)取自\([0,+\infty)\)，运行成本定义在\([t,t + T]\)（\(T>0\)）上。这类问题的最优控制用标准方法可得，但通常是时间不一致的。我们引入一种新型黎卡提方程，旨在找到时间一致、局部最优且时间不变的均衡策略。该方程生成器依赖未知量的延迟项，是带延迟的倒向常微分方程，等价于带超前项的正向常微分方程，本质是弗雷德霍姆积分方程，求解有挑战性。我们通过推导先验估计并应用勒雷 - 绍德尔不动点定理克服困难，建立了两个矩阵值非线性代数方程的比较定理，还给出了黎卡提方程解在\(T\to\infty\)时的收敛行为。
英文摘要
In this paper, we investigate a class of so-called sub-infinite horizon stochastic linear-quadratic optimal control problems, in which the initial time $t$ is arbitrarily taken from $[0,\infty)$ and the running cost is defined over $[t,t+T]$ for a given $T>0$. The optimal control of this type of problem can be obtained by standard methods; however, it is shown that the resulting optimal control is generally time-inconsistent. Thus, instead of seeking an optimal control, which is time-inconsistent, we aim to find a time-consistent, locally optimal, and time-invariant equilibrium strategy, by introducing a new and very interesting type of Riccati equation. Its main feature is that the generator depends on a delay term of the unknown. In other words, this Riccati equation is a backward ordinary differential equation (ODE) with delay, which is equivalent to a forward ODE with advanced terms. Such an equation is essentially a Fredholm integral equation, whose solvability is challenging. We overcome the difficulty by deriving a sharp a priori estimate and applying the Leray--Schauder fixed point theorem. To this end, we establish a comparison theorem between two matrix-valued nonlinear algebraic equations. The convergence behavior of the solution to the Riccati equation as $T\to\infty$ is also provided.
