SYMPLECTIC METHODS FOR OPTIMIZATION AND CONTROL
SYMPLECTIC METHODS FOR OPTIMIZATION AND CONTROL
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发表时间：
1999
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影响因子：
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通讯作者：
A. Agrachev;R. Gamkrelidze
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作者：
A. Agrachev;R. Gamkrelidze
1.辛几何的语言在当代数学的许多分支中都得到了成功的应用，但值得注意的是，辛几何最初的发展受到变分问题的很大影响。在最优控制中，庞特里亚金最大值原理的哈密顿体系起到了至关重要的作用，该体系本身就是辛几何的对象。在最优控制的进一步发展中，对凸性分析给予了优先级。虽然凸性分析和泛函分析对发展极值问题的一般理论很有帮助，但当凸性逼近失败时，它们根本不能有效地在更高近似下研究本质上的非线性问题。因此，自从最大值原理被发现以来，一直有人尝试引入几何方法来研究，尽管不像凸法和线性方法那样普遍。这些新的几何方法主要应用于获得高阶最优性条件和构造最优综合，今天我们已经有了许多巧妙的装置和漂亮的具体结果。似乎很可能会有一个通用的框架，可以统一这些几何研究的不同方向，并将最大值原理与经典变分的极值场理论结合起来。我们相信，辛几何可以为这种统一提供适当的语言，作为这种信念的理由，我们考虑这样的陈述，根据这一陈述，条件极值问题中的拉格朗日乘子流形是拉格朗日流形。这样，长期独立存在的两个“拉格朗日”对象--极值问题理论的主要对象拉格朗日乘子和辛几何的主要对象拉格朗日子流形可以统一起来。
1. The language of Symplectic geometry is successfully employed in many branches of contemporary mathematics, but it is worth to remind that the original development of Symplectic geometry was greatly influenced by variational problems. In Optimal control crucial role was plaid by the Hamiltonian system of Pontryagin’s Maximum principle, which itself is the object of Symplectic geometry. In further development of Optimal control priorities were given to Convex analysis. Though Convex and Functional analysis are very helpful in developing the general theory of Extremal problems, they are not at all effective for investigating essentially nonlinear problems in higher approximations, when the convex approximation fails. Therefore, since the discovery of the Maximum principle, there were always attempts of introducing of geometric methods of investigation, though not as universal as the Convex and Linear methods. These new geometric methods were applied mainly for obtaining optimality conditions of higher orders and constructing the optimal synthesis, and today we already have many ingenious devices and beautiful concrete results. It seems very probable that there should be a general framework which could unify different directions of these geometric investigations and merge the Maximum Principle with the theory of fields of extremals of the classical Calculus of Variations. We are convinced that the appropriate language for such unification can be provided by Symplectic geometry, and as a justification for such a conviction we consider the statement , according to which “the manifold of Lagrange multipliers in the problem of conditional extremum is a Lagrangian manifold”. Thus two “Lagrangian” objects — Lagrange multipliers, the main object of the theory of Extremal problems, and Lagrangian submanifolds, main objects of Symplectic geometry, which existed independently for a long time, can be unified.