Convexity in Hamilton-Jacobi Theory II: Envelope Representations
Convexity in Hamilton-Jacobi Theory II: Envelope Representations
复制标题
汉密尔顿-雅可比理论中的凸性 II：包络表示
DOI：
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发表时间：
2000
期刊：
影响因子：
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通讯作者：
P. Wolenski
中科院分区：
文献类型：
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作者：
R. Rockafellar;P. Wolenski
上下包络表示是为与最优控制问题相关的值函数和完全凸的变化演算而开发的，在状态和速度上都表现出凸性。这种凸性用于对偶上包络表示以获得下包络表示，这具有以前在这种一般性中未被感知到的优点，并且在某些情况下可以被视为提供，至少对于值函数，扩展的Hopf- Lax公式超出了状态独立哈密顿量的情况。
Upper and lower envelope representations are developed for value functions associated with problems of optimal control and the calculus of variations that are fully convex, in the sense of exhibiting convexity in both the state and the velocity. Such convexity is used in dualizing the upper envelope representations to get the lower ones, which have advantages not previously perceived in such generality and in some situations can be regarded as furnishing, at least for value functions, extended Hopf--Lax formulas that operate beyond the case of state-independent Hamiltonians.
The derivation of the lower envelope representations centers on a new function called the dualizing kernel, which propagates the Legendre--Fenchel envelope formula of convex analysis through the underlying dynamics. This kernel is shown to be characterized by a kind of double Hamilton--Jacobi equation and, despite overall nonsmoothness, to be smooth with respect to time and concave-convex in the primal and dual states. It furnishes a means whereby, in principle, value functions and their subgradients can be determined through optimization without having to deal with a separate, and typically much less favorable, Hamilton--Jacobi equation for each choice of the initial or terminal cost data.