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Über dieses Buch

This book showcases a subclass of hereditary systems, that is, systems with behaviour depending not only on their current state but also on their past history; it is an introduction to the mathematical theory of optimal control for stochastic difference Volterra equations of neutral type. As such, it will be of much interest to researchers interested in modelling processes in physics, mechanics, automatic regulation, economics and finance, biology, sociology and medicine for all of which such equations are very popular tools.

The text deals with problems of optimal control such as meeting given performance criteria, and stabilization, extending them to neutral stochastic difference Volterra equations. In particular, it contrasts the difference analogues of solutions to optimal control and optimal estimation problems for stochastic integral Volterra equations with optimal solutions for corresponding problems in stochastic difference Volterra equations.

Optimal Control of Stochastic Difference Volterra Equations commences with an historical introduction to the emergence of this type of equation with some additional mathematical preliminaries. It then deals with the necessary conditions for optimality in the control of the equations and constructs a feedback control scheme. The approximation of stochastic quasilinear Volterra equations with quadratic performance functionals is then considered. Optimal stabilization is discussed and the filtering problem formulated. Finally, two methods of solving the optimal control problem for partly observable linear stochastic processes, also with quadratic performance functionals, are developed.

Integrating the author’s own research within the context of the current state-of-the-art of research in difference equations, hereditary systems theory and optimal control, this book is addressed to specialists in mathematical optimal control theory and to graduate students in pure and applied mathematics and control engineering.

Inhaltsverzeichnis

Frontmatter

Chapter 1. Stochastic Difference Volterra Equations and Some Auxiliary Statements

Abstract
Here, an origin of stochastic difference Volterra equations of neutral type is described and some necessary definitions and auxiliary assertions are given.
Leonid Shaikhet

Chapter 2. Optimal Control

Abstract
In this chapter, optimal control problems for nonlinear and linear equations are considered.
Leonid Shaikhet

Chapter 3. Successive Approximations to the Optimal Control

Abstract
In this section, successive approximations to the optimal control of a quasilinear stochastic difference equation with quadratic performance functional are constructed.
Leonid Shaikhet

Chapter 4. Optimal and Quasioptimal Stabilization

Abstract
Here the problem of the optimal stabilization for a linear stochastic difference Volterra equation and quadratic performance functional is considered.
Leonid Shaikhet

Chapter 5. Optimal Estimation

Abstract
In this chapter, we consider the problem of constructing the optimal (in the mean square sense) estimate of an arbitrary partially observable Gaussian stochastic process from its observations with delay.
Leonid Shaikhet

Chapter 6. Optimal Control of Stochastic Difference Volterra Equations by Incomplete Information

Abstract
In this chapter, two methods for solution of the optimal control problem of a partially observable linear stochastic process with a quadratic performance functional are considered.
Leonid Shaikhet

Backmatter

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