论文标题
在欧几里得空间上的汉密尔顿 - 雅各比 - 贝尔曼方程的渐近分析
Asymptotic analysis for Hamilton-Jacobi-Bellman equations on Euclidean space
论文作者
论文摘要
已知在变化计算中的值函数的长期平均行为已知与相应亚伯均值的限制的存在有关。仍然在Tonelli情况下，这种限制反过来与关键汉密尔顿 - 雅各比方程的解决方案的存在有关。本文的目的是解决整个欧几里得空间上的类似问题，而哈密顿尔顿人则无法成为Tonelli。我们首先研究时间平均值函数的收敛性，因为时间范围进入无穷大，证明了通用控制系统的临界常数的存在。然后，我们表明，梯形方程允许与满足Lie代数等级条件的向量场相关的系统的解决方案。最后，我们在整个空间上构建了HJB方程的关键解，该解决方案与其Lax-Oleinik的演化一致。
The long-time average behavior of the value function in the calculus of variations is known to be connected to the existence of the limit of the corresponding Abel means. Still in the Tonelli case, such a limit is in turn related to the existence of solutions of the critical Hamilton-Jacobi equation. The goal of this paper is to address similar issues when set on the whole Euclidean space and the Hamiltonian fails to be Tonelli. We first study the convergence of the time-averaged value function as the time horizon goes to infinity, proving the existence of the critical constant for a general control system. Then, we show that the ergodic equation admits solutions for systems associated with a family of vector fields which satisfies the Lie Algebra rank condition. Finally, we construct a critical solution of the HJB equation on the whole space which coincides with its Lax-Oleinik evolution.
