Optimality principles, dynamic programming, and applied problems
Nikolai Botkin and Varvara Turova (Technische Universitat Munchen
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.