Stochastic Control Problems

The general theory of stochastic processes originated in the fundamental works of A. N. Kolmogorov and A. Ya. Khincin at the beginning of the 1930s. Kolmogorov, 1938 gave a systematic and rigorous construction of the theory of stochastic processes without aftereffects or, as it is customary to say nowadays, Markov processes. In a number of works, Khincin created the principles of the theory of so-called stationary processes.

Direct Numerical Methods for Optimal Control Problems

Development of interior point methods for linear and quadratic programming problems occurred during the 1990’s. Because of their simplicity and their convergence properties, interior point methods are attractive solvers for such problems. Moreover, extensions have been made to more general convex programming problems.

Selected Topics from Functional and Convex Analysis

Indirect Methods for Optimal Control Problems

This chapter is dedicated to the numerical approximation of Optimal Control Problems. The algorithms are based on the necessary conditions for optimality which allow us to use a descent method for the minimization of the cost functional.

A Control Problem for a Class of Epidemics

We consider a mathematical model corresponding to a class of epidemics. Controlling an epidemic is usually difficult. To implement the control policy suggested by a mathematical analysis in the real world is never easy. However, suggestions can be given to the public authorities about the effects of a particular control policy, and in this sense analysis and simulation by mathematical models becomes a powerful tool.

Optimal Control for Plate Problems

The variational approach leading to indirect methods Optimal Control Problems is applied to the study of simply supported and clamped plates. A unified approach based on distributed optimal control problems governed by second order elliptic boundary value problems and their penalization is used.

Numerical Approximation of Elliptic Variational Problems

This chapter is dedicated to the study of Elliptic Variational Inequalities (EVI). Different forms of such an EVI are considered. The Ritz—Galerkin discretization method is introduced, and methods to approximate the solution of an EVI are presented. The finite dimensional subspaces are built by use of the Finite Element Method. The discretized problems are solved using variants of the Successive OverRelaxation (SOR) method. The algorithms are tested on a typical example. The way to develop computer programs is carefully analysed.