Estimation and control of dynamical systems (Q1708167)

This book considers modeling and control problems of dynamical systems. The situation in which the input-output relationship is obtained through the evolution of a state variable is considered. A key element is the state representation of dynamical systems, also called the internal representation. Previously, dynamical systems were modeled by an external representation, also called an input-output relation. The internal representation introduces the very important concept of state of the system. The structure of these models is supposed to be known. Such models are nonlinear, but a linearization can be made. The question of internal representation from a given external representation is discussed. In the book it is claimed that an external representation can be always obtained from an internal representation. Control theory has been already applied successfully in many fields, such as economics, management science and mathematical finance. The main purpose is to develop control theory for a very large family of systems. Stochastic control problems of linear dynamical systems with full and partial information as well as of non-Markov systems are also considered. Nonlinear systems with non-quadratic payoffs are investigated. It is shown that for deterministic systems the Pontryagin's maximum principle can be used to derive necessary conditions of optimality for an optimal control. Problems of stochastic optimal control are considered as well. Applications of stochastic control algorithms to problems of financial mathematics are given. A financial market with \(n\) risky assets is considered. The price of assets is a random process. The Wiener process represents the only source of uncertainties in the financial market. Each component is an uncertainty, and the number of uncertainties is equal to the number of risky assets. These uncertainties are external and influence the evolution of the price of risky assets. The fact that the number of risky assets and the number of uncertainties are the same is a part of the assumption of complete markets. Stochastic control problems arising in this context are considered. The consumer-investor problem is considered. The methods of dynamic programming and the martingale method are applied. Many connections between estimation and control, as for example in the presentation of controllability and observability are discussed. Classical estimation methods such that the maximum likelihood and the Bayesian approach are reviewed.