Mathematical Sciences: Abnormal Minimizers and Discontinuous Value Functions in Optimal Control

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9622967 Ledzewicz The project adresses two fundamental approaches to finding the solution of optimal control problems: 1) finding the optimal processes through an analysis of the Maximum principle and 2) finding the value-function as a solution of the Hamilton-Jacobi-Bellman equation. In this project the proposer investigates new nontrivial optimality conditions for abnormal trajectories and their role in connection with discontinuities in the value function. Most existing results for optimality do not apply to abnormal extremals or require continuity of the value function. The occurence of abnormal processes is related to the fact the surjectivity condition in the Lyusternik theorem is not satisfied and as a result the classical nontrivial results cannot be derived. While mathematically desirable, continuity of the value function relates to small-time local controllability and need not be satisfied for many control systems with free terminal time. Abnormal processes which satisfy the Maximum Principle do so regardless of the objective. Hence if they are really optimal this strongly hints that in a certain sense they are the only possible candidates to solve the problem whereas closeby trajectories fail to do so. Thus optimal abnormal processes somehow correspond to limiting or boundary-like behaviors of optimal trajectories which strongly correlates them with discontinuities in the value function. It is expected that the analysis of discontinuities in the value function coupled with existing theory provides a solution methodology to the general problem in optimal control. The proposal consists of two separate but closely related parts. The first part is a continuation of previous work and addresses further developments of the proposer's earlier theory of nontrivial optimality conditions for abnormal processes based on second order approximations. A high-order generalization of the Lyusternik theorem without surjectivity condition and high -order approximations to the constraint sets are investigated. Using these results extended nontrivial first and second order necessary conditions for optimality of both normal and abnormal problems in optimization and optimal control are analyzed. A generalization of the proposer's existing results to a nonsmooth setting coupled with direct approximations in the dual space by means of normal cones is being pursued. In the second part of the project the proposer investigates sufficient conditions for the optimality of abnormal trajectories. In particular, the proposer analyses the role which is played by optimal abnormal extremals in the construction of a regular synthesis. %%% There are two fundamental approaches to finding the solution of optimal control problems: 1) finding the optimal processes through an analysis of the Maximum principle and 2) finding the value-function as a solution of the Hamilton-Jacobi-Bellman equation. Both of these are in general rather difficult objectives to achieve and depend on the specifics of the problem under investigation. On the other hand, judging by the known examples the discontinuities of the value-function seem to relate rather directly to abnormal processes. Since both abnormal processes and discontinuities of the value function are indicating the limiting behaviors of optimal trajectories the proposer expects that there exists a relation between these two phenomena in general. In this project the proposer investigates optimal abnormal trajectories in optimal control and their role in connection with discontinuities in the value function. It is expected that the analysis of discontinuities in the value function coupled with existing theory provides a solution methodology to the general problem in optimal control. The proposal consists of two separate but closely related parts. The first part is a continuation of previous work and addresses further developments of the proposer's earlier theory of o ptimlity conditions for abnormal problems based on second order approximations. Using high-order approximations the proposer generalizes these results to obtain nontrivial first and second order necessary conditions for optimality of both normal and abnormal problems in optimization and optimal control. A generalization of the proposer's existing results to a nonsmooth setting coupled with direct approximations in the dual space by means of normal cones is being pursued. In the second part of the project the proposer investigates sufficient conditions for the optimality of abnormal trajectories. In particular, the proposer analyses the role which is played by optimal abnormal extremals in the construction of a regular synthesis, which is an essential part necessary for the complete solution of the optimal control problem. ***