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

9503356 Shaettler使用场论方法的极值轨迹和奇异值函数的几何性质之间的联系将进行调查。通常，值函数中的奇异性发生在极值轨迹需要终止时，因为它们失去了最优性。 将研究竞争控制策略的共枕点和切割轨迹的关系。最优控制在割轨迹上的非唯一性是价值函数不可微的主要原因。 在这一分析的主要工具将是一个几何理论的共轭点和相应的建设领域的极值。当极值流光滑地覆盖状态空间1-1时，可以用类似于经典Hamilton-Jacobi理论的特征线方法构造极值的值函数和相应的正则综合。这种构造可以分段完成，因此直接适用于破极值。它还具有可以立即局部化的优点，并且可以获得局部最优性的Jacobi型充分条件。它的目的是发展一种理论，适用于任意分段光滑极值统一现有的概念，非奇异极值与那些砰砰的轨迹。对于非奇异极值（满足加强的Legendre条件），得到了表征共轭点的通常Riccati方程，而对于Bang-Bang控制，共轭点约化为某些切换点。 特别强调将给予最优控制系统与紧凑的控制集和一个哈密尔顿函数，这是二次的，更一般地严格凸的控制。在一般条件下，极大值原理在余切丛上生成一个分段定义的动力系统，该系统仍然具有局部存在性和唯一性，即所谓的混合系统。 %使用场论方法的极值轨迹和奇异值函数的几何性质之间的联系将进行调查。特别是，奇异值函数和共轭点/切割轨迹的竞争控制策略之间的连接将进行调查。最优控制在割轨迹上的非唯一性是价值函数不可微的主要原因。 在这一分析的主要工具将是一个几何理论的共轭点和相应的建设领域的破极值。这样做的优点是，结果可以立即局部化和Jacobi型局部最优性的充分条件可以得到。 它的目的是发展一种理论，适用于任意分段光滑极值统一现有的概念，非奇异极值与那些砰砰的轨迹。 特别强调将给予最优控制系统与紧凑的控制集和一个哈密尔顿函数，这是二次的，更一般地严格凸的控制。在这种情况下，极大值原理通常在余切丛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. ***