论文标题
第二阶的最佳控制问题条件与端点构成和凸控制构成
Second order necessary conditions for optimal control problems with endpoints-constraints and convex control-constraints
论文作者
论文摘要
在此手稿中，我们考虑了一个由普通的差分方程在Riemannian歧管上支配的控制系统，其终点满足了一些不等式和平等，并且其控制限制为封闭的凸组集。我们关注该系统的最佳控制问题，并从凸变化的意义上获得二阶必需条件（定理2.2）。为此，我们首先通过凸集的分离定理获得优化问题（定理4.2）的二阶必需条件。然后，我们通过将最佳控制问题转换为优化问题来得出必要的条件。值得指出的是，我们必要的限制会演变出曲率张量，这在欧几里得案中很微不足道。而且，即使是M也是欧几里得空间，我们的结果仍然令人感兴趣。实际上，我们给出了一个示例（示例2.1），该示例表明，当最佳控件停留在控制集的边界时，现有结果在定理2.2工作时无效。
In this manuscript, we consider a control system governed by a general ordinary differential equation on a Riemannian manifold, with its endpoints satisfying some inequalities and equalities, and its control constrained to a closed convex set. We concern on an optimal control problem of this system, and obtain the second order necessary condition in the sense of convex variation (Theorem 2.2). To this end, we first obtain a second order necessary condition of an optimization problem (Theorem 4.2) via separation theorem of convex sets. Then, we derive our necessary condition by transforming the optimal control problem into an optimization problem. It is worth to point out that, our necessary condtition evolves the curvature tensor, which is trivial in Euclidean case. Moreover, even M is a Euclidean space, our result is still of interest. Actually, we give an example (Example 2.1) which shows that, when an optimal control stays at the boundary of the control set, the existing results are invalid while Theorem 2.2 works.
