Switching in time-optimal problem
Switching in time-optimal problem
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发表时间：
2017-09
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通讯作者：
Carolina Biolo
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作者：
Carolina Biolo
本文的主要研究对象是一类仿射控制系统q = f0(q) + u1f1(q) + q的时间最优问题。+ ukfk(q), q∈M （Σ）其中f0，…， fk是k + 1个定义在流形M上的向量场。我们假设……， fk是光滑的（C∞(M)）， u = (u1，…， uk)是在k维闭酉球上取值的L∞可容许控制。本文基于（Σ）的几何局部性质与代数结构之间的关系，利用最优控制理论的经典方法：庞特里亚金极大值原理、二阶最优性条件等方法，作为系统李括号上的组态，分析了系统（Σ）的局部正则性。我们感兴趣的是找到系统（Σ）在M中点q ā的向量场上的一般条件，使得（Σ）靠近q ā的每个时间最优轨迹都是分段光滑的，具有有限数量的光滑分量，称为弧。更准确地说，我们寻找保证抖振现象不存在的一般条件，即在有限时间内存在收敛的光滑弧系列。我们在第三章和第四章中发现了这种情况。特别地，我们证明了在k = n−1的情况下，就李括号关系而言，所有最优控制都是光滑的或只有孤立的跳跃不连续；我们描述了庞特里亚金极值的流动。在第五章中，我们分析全局奇点的数目，称为开关，考虑（Σ）作为一个线性系统：线性漂移f0，常数可控向量场。我们证明在规则的时间间隔内会出现一个唯一的或无穷多个开关。最后，在第6章中，我们提出了我们能够证明在前几章中发现的破碎极值轨迹的最优性的情况。第三章：a . a . Agrachev， C. Biolo, switch in time-optimal problem with control in a ball, arXiv:1610.06755(2016)，将出现在SIAM J. control Optim上。第4章：A. A. Agrachev， C. Biolo，切换时间最优问题：基于二维控制的三维情况，动态控制系统，DOI: 10.1007/s10883-016-9342- 7,2016。第6章：a . a . Agrachev， C. Biolo，断裂极值的最优性，预印本2017。
The central object of this thesis is time-optimal problem on an affine control system of type q̇ = f0(q) + u1f1(q) + . . . + ukfk(q), q ∈ M (Σ) where f0, . . . , fk are k + 1 vector fields defined on the manifold M . We assume that f0, . . . , fk are smooth (C∞(M)) and u = (u1, . . . , uk) are L∞ admissible controls taking value in the k -dimensional closed unitary ball. We analyse the local regularity of system (Σ), with the classical methods of the optimal control theory: the Pontryagin maximum principle, the second order optimality conditions and other methods based on the relations between geometric local properties of (Σ) and algebraic structure, as the configurations on the Lie brackets of the system. We are interested in finding generic conditions on the vector fields of the system (Σ) in a point q̄ in M , such that each time-optimal trajectory of (Σ) close to q̄ is piece-wise smooth with a finite number of smooth components, called arcs. More precisely, we look for generic conditions that guaranties the absence of chattering phemonema, i.e. the existence of a convergent series of smooth arcs in finite time. We found this conditions in chapters 3 and 4. In particular, we show that in the case of k = n − 1 there are sufficient conditions in terms of Lie bracket relations for all optimal controls to be smooth or to have only isolated jump discontinuities; and we characterized the flow of Pontryagin’s extremals. In Chapter 5 we analyse the global number of singularities, called switchings, considering (Σ) as a linear system: with linear drift f0 , and constant controllable vector fields. We show that there will appear a unique or an infinity number of switchings at regular time intervals. Finally, in Chapter 6 we present the cases for which we were able to prove the optimality of the broken extremal trajectory, we found in the previous chapters. Here we list all the works collected in this thesis: Chapter 3: A. A. Agrachev, C. Biolo, Switching in time-optimal problem with control in a ball, arXiv:1610.06755 (2016), to appear on SIAM J. Control Optim. Chapter 4: A. A. Agrachev, C. Biolo, Switching in time-optimal problem: the 3-D case with 2-D control, J Dyn Control Syst, DOI 10.1007/s10883-016-9342-7, 2016. Chapter 6: A. A. Agrachev, C. Biolo, Optimality of a broken extremal, preprint 2017.