Linear programming based approaches to optimal control problems with long run average optimality criteria

The thesis aims at the development of mathematical tools for analysis and construction of near optimal solutions of optimal control problems with long run average optimality criteria (LRAOC). It consists of three parts. In Part I, we establish that near optimal controls of these problems can be constructed on the basis of solutions of semi-infinite dimensional linear programming (SILP) problems and their duals. The latter are shown to be approximations of the Hamilton-Jacobi-Bellman inequality corresponding to the LRAOC problem. In Part II, we extend the consideration of Part I to singularly perturbed LRAOC problems. Our approach to these problems is based on amalgamation of averaging and linear programming based techniques. We show that an asymptotically near optimal solution of the singularly perturbed problem can be constructed on the basis of an optimal solution of the averaged LRAOC problem and we show that the optimal solution of the latter can be found with the help of linear programming based techniques. Some of the results obtained in Parts I and II are stated in the form of algorithms, the convergence of which is discussed and which are illustrated with numerical examples. In Part III, we study families of SILP problems depending on a small parameter. The family of SILP problems is regularly (singularly) perturbed if its optimal value is continuous (discontinuous) at the zero value of the parameter. We introduce a regularity condition such that if it is fulfilled, then the family of SILP problems is regularly perturbed and if it is not fulfilled, then the family is likely to be singularly perturbed. We establish relationships between the regularity condition for SILP problems and regularity conditions used in dealing with perturbed LRAOC problems.