# Global Control Theory and Its Applications

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## Date

2015

## Journal Title

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## Publisher

Nova Science Publishers

## Abstract

This chapter presents a controlled dynamic system whose state at any moment in time
is described by the vector x. To control means that the systems equations are subdefinite.
If this subdefiniteness with fixed t;x is represented by some non fixed element u, then controlling the system implies closing its equations by assigning the function u(t) or u(t;x). The assignment of u(t) is called the program of control. It is the aim of control to guide the system in the assigned conditions as reflected in the constraints imposed on the state x at the end of the control period t = t f , to provide
process admissibility connected with fulfilment of current constraints imposed on the state x(t) fulfil some qualitative dynamic requirements set for the system, such as stability. In the process of solving these problems, some new mathematical ideas were suggested, namely , the theory of optimal control, developed as a synthesis of the classical variational calculus which makes use of Euler-Lagrange variation method. The first one is connected with generalization of conditions for a local minimum of the real variable function f (x) = 0; f (x) ≥ 0 to the problem of functional analysis. The second
one is connected with imbedding the optimal process construction problem in the family of these problems for various initial conditions. The second method is more demanding and it culminates in global control theory. Methods of control improvement are discussed including computer methods or iterative techniques of control improvement, namely, the gradient method and needle method. The new idea of the theory of optimal control makes use of the approach based on sufficient conditions for global optimality of control processes and a mathematical technique for global estimates related to it. An interesting characteristic feature of the global estimates technique is the richness of the analytical medium created by this technique. Some areas of application are provided by the control of the pure inertia plant, sliding mass, sliding regimes, and electricity generation.

## Description

Contributed chapter in Control Theory: Perspectives, Applications and Developments, edited by Francisco Miranda

## Keywords

Optimization, optimal control