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Optimality principles, dynamic programming, and applied problems

Organizers:
Nikolai Botkin and Varvara Turova (Technische Universitat Munchen Zentrum Mathematik)

Abstract:
Very often, applied problems assume optimization of some objective indexes to improve the quality of technological processes or to ensure safe performance of engines or plants. Optimality principles play the crucial role in the analysis of related optimal control problems and in the development of numerical methods. One of the famous optimality principles is Pontryagin's maximum principle based on the computation of variations of the objective functional using a technique of adjoint equations. It is remarkable that these variations can be used similar to the gradient in numerical descent methods. This idea is especially applicable to optimal control problems describing by partial differential equations. Another optimality principle is dynamic programming method proposed by R. Bellman at the end of 50s. This method became a powerful tool of modern control theory. It is applicable to a wide class of dynamical systems with state constraints and uncertain factors interpreted as counteractions of an opponent. In problems with continuous time, dynamic programming approach leads to Hamilton- Jacobi-Bellman (HJB) equations whose theory has been intensively developed recently. Nowadays, the development of effective numerical methods is of great importance for applications. It should be noticed that enhanced methods of characteristics become an interesting tool in the analysis and numerics for HBJ. The section focuses on recent achievements in the development of optimality principles and translation of them into numerical algorithms. Some important applications related to optimal control problems of complex radiative-conductive heat transfer and inverse problems of heat exchange will be presented. Methods of constructing viscosity solutions of HJB equations arising from nonlinear control problems will be discussed. Various important applications of dynamic programming method in different areas will be demonstrated.