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. ***
