Vanishing Discount Limit and Nonexpansive Optimal Control and Differential Games
Vanishing Discount Limit and Nonexpansive Optimal Control and Differential Games
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DOI：
10.1137/130945429
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发表时间：
2015-07
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影响因子：
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通讯作者：
P. Cannarsa;M. Quincampoix
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文献类型：
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作者：
P. Cannarsa;M. Quincampoix
当折现因子$\ λ $趋于零时，研究具有无限视界的折现代价泛函的最优值$V_\ λ $的极限行为是遍历控制中的一个经典问题。在文献中，这个问题已经在各种条件下解决了，以确保重新缩放的值函数$\lambda V_\lambda$一致收敛到一个常数极限。本文的主要目的是在没有这些条件的情况下研究这个问题，从而使上述极限不必是常数。因此，在非扩张性假设下，我们导出了Lipschitz界，使得控制系统和微分对策的紧性都为$\{\ λ V_\ λ \}$。然后，我们研究了Hamilton- Jacobi方程在哈密顿量是径向非递减的假设下解的收敛性，从而允许存在非矫顽力方向。利用PDE方法，我们证明了收敛性是单调的，并将极限描述为一个正则表达式的极大子解。
A classical problem in ergodic control consists of studying the limit behavior of the optimal value $V_\lambda$ of a discounted cost functional with infinite horizon as the discount factor $\lambda$ tends to zero. In the literature, this problem has been addressed under various conditions ensuring that the rescaled value function $\lambda V_\lambda$ converges uniformly to a constant limit. The main goal of this paper is to study this problem without such conditions, so that the aforementioned limit need not be constant. So, under a nonexpansivity assumption, we derive Lipschitz bounds which yield compactness of $\{\lambda V_\lambda\}$ for both control systems and differential games. Then, we study the convergence of solutions to Hamilton--Jacobi equations under the hypothesis that the Hamiltonian is radially nondecreasing, hence allowing for the existence noncoercivity directions. Using PDE methods, we show that the convergence is monotone and we characterize the limit as the maximal subsolution of a cert...