Boundary Control Problem of Infinite Order Distributed Hyperbolic Systems Involving Time Lags

Various optimal boundary control problems for linear infinite order distributed hyperbolic systems involving constant time lags are considered. Constraints on controls are imposed. Necessary and sufficient optimality conditions for the Neumann problem with the quadratic performance functional are derived.


Introduction
Distributed parameters systems with delays can be used to describe many phenomena in the real world.As is well known, heat conduction, properties of elastic-plastic material, fluid dynamics, diffusion-reaction processes, the transmission of the signals at a certain distance by using electric long lines, etc., all lie within this area.The object that we are studying (temperature, displacement, concentration, velocity, etc.) is usually referred to as the state.
The optimal control problems of second order distributed parabolic and hyperbolic systems involving time lags appearing in the boundary condition have been widely discussed in many papers and monographs.A fundamental study of such problems is given by [1] and was next developed by [2] and [3].It was also intensively investigated by [4][5][6][7][8][9][10][11][12][13][14] and [15,16] in which linear quadratic problem for parabolic and hyperbolic systems with time delays given in the different form (constant time delays, time-varying delays, time delays given in the integral form, etc.) were presented.
In this paper, we consider the optimal control for infinite order hyperbolic systems and for (n × n) infinite order hyperbolic systems involving constant time lags appearing in both in the state equation and in the boundary condition.Such an infinite order hyperbolic system can be treated as a generalization of the mathematical model for a plasma control process.
The quadratic performance functional defined over a fixed time horizon are taken and some constraints are imposed on the boundary control.Following a line of the Lions scheme, necessary and sufficient optimality conditions for the Neumann problem applied to the above system were derived.The optimal control is characterized by the adjoint equations.
This paper is organized as follows.In Section 1, we introduce spaces of functions of infinite order.In Section 2, we formulate the mixed Neumann problem for infinite order hyperbolic systems involving constant time lags.In Section 3, the boundary optimal control problem for this case is formulated, then we give the necessary and sufficient conditions for the control to be an optimal.In Section 4, we concluded and generalized our results.

Sobolev Spaces with Infinite Order
The object of this section is to give the definition of some function spaces of infinite order, and the chains of the constructed spaces which will be used later.
Let  be a bounded opesn set of with a smooth boundary n R  , which is a -manifold of dimension Locally,  is totally on one side of  .We define the infinite order Sobolev space of infinite order of periodic functions   x  defined on  [17][18][19] as follows:  is the space of infinite differentiable functions,  is a numerical sequence and 0 a  , , , is a vector function and The space W a  is defined as the formal conjugate space to the space , namely: From above, is everywhere dense in with topological inclusions and denotes the topological dual space with respect to , so we have the following chain of inclusions: We now introduce which we shall denoted by denotes the space of measurable functions such that In the same manner we define the spaces as its formal conjugate.Also, we have the following chain of inclusions: with norm defined by: 2 2 0, ; , 2 0, ; , The construction of the Cartesian product of n-times to the above Hilbert spaces can be construct, for example Finally, we have the following chain of inclusions: where . The spaces considered in paper are assumed to be re

Mixed Neumann Problem for Infinite me
The object of this section is to formulate the following 2 this al.

Order Hyperbolic System Involving Ti Lags
mixed initial boundary value Neumann problem for infinite order hyperbolic system involving time lags which defines the state of the system model. where has the same properties as in Section e 1.We hav  T is a specified positive number representing a finite positive number representing a time The hyperb r that in the state Equation ( 1) is an infinite order hyperbolic operator and

 
A t [19] is given by: Proof.It is well known that th Equations ( 1)-( 6) constitute a Neumann problem.Then the left-hand side of the boundary condition (5) m written in the following form:  and π t y .
We shall ormu (10) ent conditions for the exist he mixed boundary value problem (1)- (6) ( ) 0, ; , 2 0, ; which is a Hilbert space normed by where the space 0, ; W T L  denotes the So space of second order of functions defined on bolev  

0,T and taking values in
The existence of a unique solution for the mixed initial-boundary value problem ( 1)-( 6) on the cylinder Q ng a constructiv nd in il the e covers the whole cy can be proved usi e method, i.e., solving at first Equations ( 1)-( 6) on the sub-cylinder Q 1 a turn on Q 2 etc., unt procedur linder Q.In this way, the solution in the previous step determines the next one.
Then, there exists eorems Now, we formulate the optimal ol problem 1)-( 6) in the context of the Theore is

Problem Formulation and Optimization Th
The time horizon T is fixed in our problem.
The performan ls.
functional is given by: ce

  y v
We note from Theorem 2.1 optimization problem ( 1)-( 6), ( 15), (16).The solving of W of Mak the Loins's scheme we shall derive the necessary and sufficient conditions of optimality for the the formulated optim ntrol problem is equivalent to seeking a From the Lion's scheme [21] (Theorem 1.3 of, p. 10), it follows that for 2 > 0   a unique optimal control v  exists.Moreover, v  is characterized by the following condition: For the performance functional of form (15) the relation ( 17) can be expressed as ad In order to simplify (18), we introduce the adjoint equation, and for every , we define the adjoint variable as the solution of the equations: where for the adjoint problem (19)-( 24).We simplify (18) using the adjoint equation ( 19)- (24).For this purpose de for noting by (19), (20) by 19)-( 24), multiplying both sides of (v) -y(v * ), rating over and then adding both sides of ( 19), (20), we , ; , ; Using the Equation ( 1), the second integral on the right-hand side of (26) can be written as , ; , ; Using Green's formula, the third integral on t hand side of (26) can be written as Using the boundary condition (5), one can transform the second integral on the right-hand side of (28) into the form: The last component in (28) can be rewritten as 28) into (26), we obtain Substituting ( 29) and ( 30) into (28), and then (2

T h T T h A T T h A T y x t v t p x t h v c x t h y x t v y x t v t p x t v v v t p v y v y v t p v y v y v t p x t v v v t
Substituting ( 31) into (18) gives The foregoing result is now summarized.Theorem 3.1.For the problem (1)-( 6), with the performance functi d and 2 > 0 and with conditions ere exists a unique optimal control v  maxi (32).

Mathematical Examples
Example 3.1.Consider now the particular case where 16), th which satisfies the mum condition

Exampl
can also consider an analogous optimal control problem where the performance functional is given by:  .9), for each ,1 Then, the optimal control v is characterized by: We define the adjoint variable as the solution of the equations: As in the above section, we have the following result.Using the adjoint Equations ( 35)-(40) in this case, the condition (34) can also be written in the following form The following result is now summarized.), and with adjoint Equations a unique optimal control v  which satisfies the maxim Example 3.3.Case: u an analogous optimal control problem where the performance functional is given by: . We can also consider where From Theorem 2.1 and the Trace Theorem p. 9), for each [20] , there exists a unique solution . Thus, I is well defined.Then, the y optimal control u  is characterized by We define the adjoint variable     = = , ; p p u p x t u as the solution of the equations: 49) in this c condition (43) can also be written in the following form:

A p u x t c x t h p x t h u x t T h
The following result is now summarized.Theorem 3.3.For the problem (1)-( 6), ( 44)-(49 As in the abo , we have the following result.Let the hypothesis of Theorem 2.1 be satisfied.Then, f given to th Using the adjoint equations ( 4p u ). ase, the ), ( 16) with , there exists a unique o which satisfies the maximum condition (50).

Generalization
The optimal control problems presented her can be extended to certain differe rol for 2 2  coupled infinite orde s involving constant time lags.Cas rol for n n  coupled infinite order hyperbolic systems involving constant time lags.Such extension can be applied to solving many control pro nical engineering.We will extend the discussions to study the optimal ntrol for 2 2   coupled infinite order hyperbolic sys-co fu tems involving constant time lags.We consider the case where , the perf is given by: The following results can now be proved.Theorem 4.1.Let 0 y , 1 y , 0  , 0  , v and u be given with for the following mixed ini-Then, there exis l-boundary value problem: where , ; = , ; , y y x t v y x t v y x t  for the adjoint problem:

v a D p v t b x t h p x t h v p v y v z x t T h p v a D p v b x t t p x t h v p v y v z x t T h
,   = 0, = 0, ,

A c x t h p x t h v x t T h p x t c x t h p x t h v x t T h
, .
is the adjoint state.
result is now summarized.
For the pr performance function (51) with Case 2: Optimal control for n × n coupled infinite order hyperbolic systems involving constant time lags.
We will extend the discussion to n × n coupled infinite order hyperbolic systems involving con lags.We consider the case where 3), then there which satisfies the maximum co stant time , the performance functional is given by (El-Saify, 2005; 2006): where , , , . The following results can now be proved.0 , v and u be gi , , , where , ; , , ; , , , ; which satisfies the maximum condition (79).
In the case of performance functionals (15, 33, 42, 51 and 65) with 1 > 0 and 2 = 0   , the optimal control problem reduces to minimization of the functional on a closed and convex subset in a Hilbert space.Then, the optimization problem is equivalent to a quadratic pro-gramming one, which can be solved by the use of the well-known Gilbert algorithm.
In this paper, we have considered the boundary control or infinite order hyperbolic system and also for finite order hyperbolic systems involving constat time lags appearing both in the state equations the Neumann boundary conditions.We can also co ry optimal contro for   conditions.Also it is evident that by modifying: ns, (Dirichlet, Neumann, mixed, etc.);  The nature of the control (distributed, boundary, etc.)  The nature of the observation (distributed, boundary etc.); system; The time delays (constant time delays, time-varying delays, multiple time-varying delays, time delays in the integral form, etc.);  The number of variables (finite number of variables, infinite number of variables systems, etc.);  The type of equation (elliptic, parabolic, hyperbolic, etc.);  The order of equation (second order, Schrödinger, infinite order, etc.);  The type of control (o optimal control problem on the above problem ar help of [21] and Dubovitskii-Milyutin formalisms [23][24][25][26][27][28][29][30][31][32].Those problems need further i and form tasks for future research.Thes tioned above will be developed in forthcoming papers. nite

Lemma 4 . 1 .
Let the hypothesis of Theore satisfied.Then for given

|
hyp rbolic systems with timevarying delays appearing in th ons and in the Neumann or Dirichlet bound itions.We can also co For this purpose, we may apply Theorem 2.1 (with an obvious change of variables).Hence, using Theorem 2.1, the followi result can be proved.