Problems in the classical calculus of variations

This dissertation concerns convergence analysis for nonparametric problems in the calculus of variations and sufficient conditions for weak local minimizer of a functional for both nonparametric and parametric problems. Newton's method in infinite-dimensional space is proved to be well-defined and converges quadratically to a weak local minimizer of a functional subject to certain boundary conditions. Sufficient conditions for global converges are proposed and a well-defined algorithm based on those conditions is presented and proved to converge. Finite element discretization is employed to achieve an implementable line-search-based quasi-Newton algorithm and a proof of convergence of the discretization of the algorithm is included. This work also proposes sufficient conditions for weak local minimizer without using the language of conjugate points. The form of new conditions is consistent with the ones in finite-dimensional case. It is believed that the new form of sufficient conditions will lead to simpler approaches to verify an extremal as local minimizer for well-known problems in calculus of variations.

GENERATING PLANS IN CONCURRENT, PROBABILISTIC, OVER-SUBSCRIBED DOMAINS

Planning in realistic domains typically involves reasoning under uncertainty, operating under time and resource constraints, and finding the optimal subset of goals to work on. Creating optimal plans that consider all of these features is a computationally complex, challenging problem. This dissertation develops an AO* search based planner named CPOAO* (Concurrent, Probabilistic, Over-subscription AO*) which incorporates durative actions, time and resource constraints, concurrent execution, over-subscribed goals, and probabilistic actions. To handle concurrent actions, action combinations rather than individual actions are taken as plan steps. Plan optimization is explored by adding two novel aspects to plans. First, parallel steps that serve the same goal are used to increase the plan’s probability of success. Traditionally, only parallel steps that serve different goals are used to reduce plan execution time. Second, actions that are executing but are no longer useful can be terminated to save resources and time. Conventional planners assume that all actions that were started will be carried out to completion. To reduce the size of the search space, several domain independent heuristic functions and pruning techniques were developed. The key ideas are to exploit dominance relations for candidate action sets and to develop relaxed planning graphs to estimate the expected rewards of states. This thesis contributes (1) an AO* based planner to generate parallel plans, (2) domain independent heuristics to increase planner efficiency, and (3) the ability to execute redundant actions and to terminate useless actions to increase plan efficiency.