الفهرس 
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Abstract
Optimal control theory has been used in the solution of an enormous variety of problems in physics, engineering, economic and biology. Many of the problem of design in airframe, shipbuilding, electronic, and other engineering fields are in essence, problems of control.
To define a classical control problem, we require to describe the components of the problem. To start the definitions we need
(i) a real closed time interval I = [t_0; T]; with t_0< T;
(ii) a bounded and closed subset U of R^n that in which the control functions take values,
(iii) a differential equation describing the control system, satisfied by the trajectory function t ∈ I → y(t) ∈R^n; and a control function t ∈ I → y(t) ∈R^n;
(iv) an observation function z(t; y(t); u(t)) which is assumed to be known.
We can put further conditions on this function as necessary.
A classical optimal control problem is of finding an admissible control u ∈ U which satisfies the differential equation describing the controlled system and minimizes the cost functional J(y; u) =∫_I▒〖f0(t; u; y)dt.〗 The objective is to determine an admissible control, called optimal control, that provides a satisfactory state for us and that minimizes the value of functional J .
Perhaps the most widely studied type of problem in the mathematical theory of control is the ” time optimal ” control problem. The aim of control in this problem is to transform(or steering) any initial state of a dynamic system into a desired stationary state (or to hitting a target set)in minimal time. i.e., the cost function in this problem is the time in which a system is driving to desired state. Such problems are physical meaningful only if constraints are imposed on the control variables u; since otherwise, trivially, the desired state could be achieved in zero time by the application
of controls of infinite amplitude.
The solution of the time-optimal control problem subject to bounded input is bang-bang control, i.e., the control in which the input variable takes either the maximum or minimum values. We mention the work of Wang[59], where a bang-bang principle of time optimal internal controls of heat equation was considered.
Time-optimal control has been studied extensively for ”lumped parameter systems” i.e., systems governed by ordinary differential equations. We refer the readers to classical books for Henry Herms and Joseph P.Lasalle [32] and Knowles[36]
In recent years, significant emphasis has been given to study the optimal control for systems described by partial differential equations, here referred to as distributed systems, arises whenever the spatial distribution of variables is to be controlled. Various optimization problems associated with the optimal control of distributed parameter systems have been studied recently by many authors, we mention only, Fattorini [24]-[30], Wang [57], Barbu [8], and others in, [60], [44], [33], [7]-[6],[13]-[16],[2], [4] besides
the definite book ”Optimal control of system governed by partial differential equations” written by J.L.Lions [45] which is itself based on his monumental joint work with Magenes [46]. In our M.Sc. thesis [52],[50]-[51] and [17]-[39], we discussed some time optimal control problems for some parabolic systems.
In practical applications, the behavior of many dynamical systems which describes a state of time-optimal control problems depends upon their past histories. This phenomenon can be induced by the presence of time delays. Due to the inherent difficulties in solving control problems with time delays, the progress in this area has been slow. Here, we mention the work of Wang [58], where the time optimal control for a class of ordinary differential-difference equation with time lag was considered. Also, we mention the work of Knowles [35], where a Time optimal control of parabolic systems with boundary condition involving time delays was considered and it is shown that the optimal control is characterized in terms of an adjoint system and it is of the bang-bang type.
Time-optimal control of distributed parameter systems governed by a system of hyperbolic equations is of special importance for the active control of structural systems for which the equations of motion are generally expressed by hyperbolic differential equations. A typical application of a hyperbolic equation is the vibrating system:
Let y∶ [0; T]→ H, for any T > 0; be a function that describes the deviation of a vibrating medium from the position of rest as a function of time t with values in a Hilbert space H. We assume y to satisfy an abstract wave equation of the form
y^’’ (t) + Ay(t) = u(t) (1)
where y^’ denotes the derivative with respect to t; A is a self adjoint positive definite linear operator defined on a dense domain D(A) in H and u(t) ∈ H for almost all t ∈ [0; T],〖||.|| 〗_H is measurable and satisfies ∫_0^t▒〖〖||u(t)||〗_H^2 dt〗 <∞
where 〖||.|| 〗_H denotes the norm in H. The space of all (classes of) such functions is called L2([0; T];H).
In [21] and [40] the following time-optimal control problem was investigated: Let y_0,y_1,(y_0 ) ̅,(y_1 ) ̅ ∈ H,M > 0: Does there exist a time T > 0 and a control function u ∈ L2([0; T];H) with ∫_0^t▒〖〖||u(t)||〗_H^2 dt <M〗 such that the corresponding solution of Equation (1) with y(0)= y_0,y^’ (0) = y_1: satisfies y(T)= (y_0 ) ̅,y^’ (T) = (y_1 ) ̅.
The results in [21] partly overlap with results in [40] and they were shown that : For every T > 0 there exists exactly one control function of the above problem and this control is bang-bang i.e 〖||u(t)||〗_H^ = M.
Numerical methods for solving minimum-time optimal control problem of the inhomogeneous wave equation has been presented in works such as [42]- [41].
In this dissertation, the time-optimal control problem for a regular hyperbolic system ( such as wave equation in [21] and [40]) has been extended
in the following different directions :
1. co-operative systems (coupled and n x n system )
2. several cases of observations
3. hyperbolic systems having delay
4. boundary controls
5. new systems which solution belong a new sobolev spaces
6. control-state constraints
as follows:
(1) In Chapter 2, for the following cases observation
(a) position observation y(t);
(b) velocity observation y0(t);
(c) position-velocity observation (y(t); y0(t));
the time-optimal control problem of 2 x 2 co-operative hyperbolic system with variable coefficients involving Laplace operator has been considered. The optimality condition of a time-optimal control has been obtained in the form of maximum principal and in particular case of the space of controls, maximum principal has been used to derive specific properties of the optimal control, such as bang - bangness and uniqueness, also the results have been extended to n x n systems.
(2) In Chapter 3, the time-optimal control problem for co-operative hyperbolic system with boundary controls and position observation has been considered. First, the existence of a unique solution of the above hyperbolic system with Neumann boundary conditions has been defined by using Transposition Theorem, then, necessary conditions for the time-optimal control of hyperbolic systems with Neumann boundary control have been derived in the form of Maximum principle. Finally, the time-optimal control of hyperbolic systems with Dirichlet boundary control has been considered.
(3) In Chapter 4, the time-optimal control problem for co-operative hyperbolic systems with time-delay and position observation has been considered. First, the existence and uniqueness of solutions for 2 x 2 co-operative hyperbolic system with time delay has been proved using constructive method. Then, the time-optimal control problem with position observation was characterized by the maximum principle. Finally, using this characterization, the bang-bang principle and the approximately controllability conditions were investigated.
(4) In Chapter 5, the results in (3) have been extended to the time optimal control problems for a new n x n infinite order hyperbolic systems. First, we have proved that this system has a unique solution in a new Sobolev spaces with infinite order. Then in the case of position observation, necessary conditions for the time-optimal controls of infinite order hyperbolic system have been derived in the form of Maximum principle.
(5) In Chapter 6, as application of Dubovitskii- Milyutin method, necessary optimality conditions for time optimal control problems of second order evolution systems with control-state constraint have been established.
(6) In Chapter 7, the time optimal control problems for a new n x n system involving Petrowisky type operator of order 2` with control-state and infinite number of variable has been studied. First, we have proved that this system has a unique solution in a new Sobolev spaces with infinite number of variables. Then in the case of position observation and velocity observation, necessary conditions for the time-optimal controls of Petrowsky system have been obtained.
(7) In Chapter 8, the time-optimal control problems for n x n system of nonhomogeneous Neumann hyperbolic type with control-state and infinite number of variable has been studied. First, we have proved that this system have a unique solution in a new sobolev spaces with infinite number of variables. Then in the case of position observation, necessary conditions for the time-optimal controls of such problem have been obtained.
Most of the results in this dissertation are published, as follows:
Results presented in Chapter 2 have been published in the following journal [20]:
El-Saify, H.A., Serag.H.M and Shehata,M.A. Time-optimal control for co-operative hyperbolic systems involving Laplace operator . Journal of Dynamical and Control systems 15,3 . 405- 423 (2009).
Results presented in Chapter 6 and 7 have been published in the following journal [18]:
El-Saify , H.A. and Shehata,M.A. Time-optimal control of Petrowsky systems
with infinitely many variables and control-state constraints Studies in Mathe-
matical Sciences Vol. 2, No. 1, pp. 21-35, (2011).
Results presented in Chapter 8 have been published in the following journal [19]:
El-Saify , H.A. and Shehata,M.A. Time-optimal boundary control of Neumann hyperbolic systems with infinitely variables and state constraints Journal of Mathematics and Computing System Vol. 2, No. 2, pp. 69-79, (2011).