Mathematical Sciences: Geometric Properties of Extremal Trajectories and Singularities of the Value Function

用场理论的方法来研究极值轨迹的几何性质和值函数的奇点之间的联系。通常，值函数中的奇点出现在极值轨迹需要终止时，因为它们失去了最优性。研究了竞争控制策略与共轭点和切迹的关系。切点上最优控制的非唯一性是值函数不可微的主要原因。这个分析的主要工具是共轭点的几何理论和相应的极值场的构造。当极值流平滑地覆盖状态空间1-1时，可以用类似经典Hamilton-Jacobi理论的特征方法构造值函数和相应的正则极值综合。这种构造可以分段完成，并直接应用于破碎的极值。它还具有可以立即局部化的优点，并且对于破极值可以获得局部最优性的Jacobi型充分条件。目的是建立一种适用于任意分段光滑极值的理论，统一现有的非奇异极值与砰砰轨迹的概念。对于满足强化Legendre条件的非奇异极值，得到了共轭点的常用Riccati方程，而对于bang-bang控制，共轭点简化为若干开关点。我们将特别强调具有紧控制集和二次哈密顿函数的最优控制系统，在控制中哈密顿函数通常是严格凸的。在一般条件下，极大原理在余切束上生成一个分段定义的动力系统，该系统仍然保持局部存在唯一性，即所谓的混合系统。用场理论的方法来研究极值轨迹的几何性质和值函数的奇点之间的联系。特别地，将研究值函数的奇异点与竞争控制策略的共轭点/切位点之间的联系。切点上最优控制的非唯一性是值函数不可微的主要原因。这种分析的主要工具是共轭点的几何理论和相应的破极值域的构造。这样做的优点是结果可以立即局部化，并且可以获得局部最优性的Jacobi型充分条件。目的是建立一种适用于任意分段光滑极值的理论，统一现有的非奇异极值与砰砰轨迹的概念。我们将特别强调具有紧控制集和二次哈密顿函数的最优控制系统，在控制中哈密顿函数通常是严格凸的。对于这种情况，极大原理通常会在余切束上生成一个分段定义的动力系统，该系统仍然保持局部存在性和唯一性，即所谓的混合系统。＊＊＊

9503356 Shaettler Using a field theoretic approach the links between geometric properties of extremal trajectories and singularities of the value function will be investigated. Typically singularities in the value-function occur as extremal trajectories need to be terminated because they lose optimality. The relation to conjugate points and cut-loci of competing control strategies will be investigated. The nonuniqueness of optimal controls on the cut-locus is the primary source for nondifferentiability of the value function. The main tool in this analysis will be a geometric theory of conjugate points and corresponding construction of a field of extremals. When the flow of extremals covers the state space 1-1 and smoothly, the value function and corresponding regular synthesis of extremals can be constructed by the method of characteristics analogous to classical Hamilton-Jacobi theory. This construction can be done piecewise and as such directly applies to broken extremals. It also has the advantage that it can immediately be localized and Jacobi type sufficient conditions for local optimality can be attained for broken extremals. It is intended to develop a theory which applies to arbitrary piecewise smooth extremals unifying existing concepts for nonsingular extremals with those for bang-bang trajectories. For nonsingular extremals (which satisfy the strengthened Legendre condition) the usual Riccati equations for characterizations of the conjugate points are obtained while for bang-bang controls the conjugate points reduce to certain switching points. Special emphasis will be given to optimal control systems with a compact control set and a Hamiltonian function which is quadratic, more generally strictly convex in the control. Under generic conditions the Maximum principle generates a piecewise defined dynamical system on the cotangent bundle for which still local existence and uniqueness properties hold, a so-called hybrid system. %%% Using a field theoretic approach the links between geometric properties of extremal trajectories and singularities of the value function will be investigated. In particular, the connections between singularities in the value function and conjugate points/cut-loci of competing control strategies will be investigated. The nonuniqueness of optimal controls on the cut-locus is the primary source for nondifferentiability of the value function. The main tool in this analysis will be a geometric theory of conjugate points and corresponding construction of a field of broken extremals. This has the advantage that results can immediately be localized and Jacobi type sufficient conditions for local optimality can be attained. It is intended to develop a theory which applies to arbitrary piecewise smooth extremals unifying existing concepts for nonsingular extremals with those for bang-bang trajectories. Special emphasis will be given to optimal control systems with a compact control set and a Hamiltonian function which is quadratic, more generally strictly convex in the control. For this case, the Maximum principle typically generates a piecewise defined dynamical system on the cotangent bundle for which still local existence and uniqueness properties hold, a so-called hybrid system. ***