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Abstract
The aim of this thesis is to study some issues of optimal control of time and find a group of variants that describe the control systems are equivalent consisting of n of partial differential equations equivalent and associated with, so these systems are governed by influences differential partial representation matrix contains the effective differential part of the second level is the number of final of the influential variables such as Laplace or differential influential part of the second order with an infinite number of variables or differential influential part of the infinite level. Finally, using the theory of Diovensk - Miluotn got the necessary conditions for optimal control for n × n systems that contain the equivalent of partial differential effects of the infinite level or effects of P 2 N with infinite number of variables. Note that in each case Consider first the existence and oneness to resolve previous regimes.
Introduction
The most important types of studies in mathematical theories of control, ”the issue of optimal control for a time of” The simplest example of this issue is the transfer of particle or point of the state primary to state definitively or to a particular goal in the shortest possible time and this issue has been studied recently for systems equivalent containing psychotropic differential partly by some and in particular singled them Lions [94], which dealt with the issue of optimal control systems are equivalent. He studied the existence of optimal control of the system and also studied some properties of this control has been given [40] Fattorini some observations on the issue of optimal control of time in systems that contain effects partial differential. The Arada and Raymond [3] examine the question of optimal control of time to a quasi-equivalent in the case of fuel restrictions on the objective function and by the controller alone.
It is worth mentioning that the time delay is an important phenomenon in industrial processes, economic systems and vitality. For example, show that the phenomenon in transport and communication. As the time delay has major impacts on the stability of dynamic systems such as this was important to include in the mathematical description, and display differential equations containing time-delayed. He has Knowles [53] examine the question of optimal control system for the equivalent time delay appears in marginal conditions. Was developed by Kowalewski and Karakowiak [76] this issue was, they studied the issue of optimal control system for the equivalent time delay appears in both the equivalent of the equation and boundary conditions at the same time. Applying the theory of Diovensk - Miluotn has Kotarski [58] Bi order to find the necessary condition ”for the optimal control of time” for the equivalent system contains an influential part differential on the ID vacuum with a number of variables is very. Finally, our results Pena position of the results described above and in the end of the offer submitted what has been achieved in each section separately.
Part One
Is the door of a preliminary nature and includes preliminary concepts and definitions and the basic theories used in this letter. It consists of three parts. The first part included the definitions and properties of spaces with deltoid configuration string of these spaces. In Part II we offer a theory of existence and oneness to solve systems Almkaqip. In the third part we show the theory Diovensk - Miluotn.
Part II
In this section, we study the ”Question of the Optimal Control of Time” to an equivalent system. This section consists of three parts. In the first part we study the issue ”final status” In the second part we study the issue in the case of the existence of ”objective function” and in both cases, we study the existence of optimal control and some of its properties (Albang Bang - the oneness of Allah). In Part III are subjected to some examples of mathematical systems equivalent (n × n systems equivalent), which contains the different types of effects (Laplace influential - influential differential with an infinite number of variables - differential effective infinite rank).
Part III
In this section we study some issues of time optimal control for n × n systems of equations equivalent to showing the time delay in the equations and boundary conditions at the same time. And This chapter consists of two parts the first part where we study the optimal control of time, ”the system of n × n of equations equivalent with variables expired, which shows the time delay. First, we describe this system and then studying the existence and oneness to solve this system at the end of this section, we study the various issues of control to the system and some properties of this control.
In the second part extends the results to discuss ”the issue of optimal control of time for a system of” n × n equations of the equivalent of a number of variables is very apparent that the time delay variable in the equations in the equivalent boundary conditions at the same time.
Part IV
And apply the theory Diovensk - Miluotn the question of optimal control of time. This section consists of two parts. In the first part we apply this theory to find the necessary condition ”for the optimal control of time” for the n × n system of equations with the equivalent rank of the infinite in the presence of restrictions on the control and also on the system. In the second part we apply the theory to find the necessary condition ”to control the optimal time of” the system of n × n of equations equivalent to showing the time delay and contain the effects partial differential P 2 N defined on spaces with a number Unfinished variables in the presence of restrictions on the control.