Convergence of The Discrete Classical Optimal Control Problem To The Continuous Classical Optimal Control Problem Including A Nonlinear Hyperbolic P.D. Equation

Our focus in this paper is to study the behaviour in the limit of the discrete classical optimal control problem including partial differential equations of nonlinear hyperbolic type. We study that the discrete state and its discrete derivative are stable in Hilbert spaces 1( ) 0 H W and 2( ) L W respectively. The discrete state equations containing discrete controls converge to the continuous state equ ations. The convergent of a subseq uence of t he sequ ence of discrete classical optimal for the discrete optimal control problem, to a continuous classical optimal c ontrol for the continuous optimal co ntrol problem is proved. Finally the necessary c onditions for optimality of t he discrete classical optimal c ontrol problem converge t o the ne cessary conditions for optimality of the continuous optimal co ntrol problem, s o as the minimum principle in blockwise form for optimality.


Introduction
The behaviour of the discrete classical optimal control problem is very important.Since it tell us that the discrete form for the continuous classical optimal control is suitable to use this discrete form or not.This importance leaded many researchers have studied about the behaviour in the limit of discrete optimal control problem involving ordinary differential equations as in [ 1], [ 2 ], [ 3 ], and [4 ].For these causes we deal with in this work the study of the behaviour for the discrete classical optimal control problem.i.e. we prove that the discrete classical optimal control problem of a nonlinear hyperbolic partial differential equations which is studied in [5] converges in the limit to the continuous classical optimal control problem of a nonlinear hyperbolic partial differential equations which is studied in [6].Therefore we describe in the first two sections some forms, assumptions, and results which were obtained from the study of the continuous and the discrete classical optimal control problem of a nonlinear hyperbolic partial differential equations.We study first the stability of discrete state in the Hilbert space 1 () 0 H Ω and its discrete derivative in the Hilbert space 2 () L Ω .Then we show that the discrete solution of the weak form state equations (in the discrete optimal control problem) converges in the limit to the solution of the weak form state equation (in the continuous optimal control problem).We prove that a subsequence of the sequence of discrete classical optimal control problem (for the discrete control problem) converges in the limit to a classical optimal control for the continuous control problem.Also we prove if a subsequence of the sequence of discrete admissible classical controls which satisfy the necessary conditions for optimality for the discrete optimal control problem then the limit of this subsequence is an admissible classical control which satisfies the necessary conditions for optimality for the continuous optimal control problem, so as the minimum principle in blockwise form for optimality.

Description of the Continuous Classical
Optimal Control Problem:-In this section, some forms, and results (which will need their in this paper) of the continuous classical optimal control problem of a nonlinear hyperbolic partial differential equations (CCOCP) are described which studied by [6].We begin with the weak form of the continuous state equations of a nonlinear partial differential equations is ( ) The continuous classical optimal control problem (CCOCP) is to finduW ∈ , such that 00 ()min() where A W ,is the set of admissible control which is defined by: where yy u = is the solution of (1-3), for the control u .This solution proved exist and unique [6].The existence of a continuous classical optimal control proved in [6] The existence of a discrete classical optimal control is obtained under the above assumptions, with

Stability:-
In this section we study the stability of the discrete state solutions and its discrete derivatives for the discrete state equations in weak form by the following lemma.Lemma 3.1:-For every discrete control nn uW ∈ , if t ∆ is sufficiently small, then

ctNcTc ≤∆==
Now, by substituting lN = , in both sides of (23), using the last results, we obtain that all the terms in the R.H.S. of (23) are bounded, and with / tbc ∆< , the 1 st , 2 nd , and the 3 rd terms in the L.H.S. of (23), become positive, and we get

Convergence:-
In this section we study the behavior of the discrete classical optimal control problem in the limit, i.e. we study the discrete classical optimal control problem and its main results which were considered in section 3 of this paper converge to the continuous classical optimal control problem and its main results were considered in section 2 of this paper.First we state the following control approximation lemma.Integrating both sides of this equality from 0 t = , to tT = , then using integrating by parts for the term in the L.H.S. of the obtained equation, we get 0 (,)() Integrating by parts the 1 st term in L.H.S. of (30), we have Gu ,(for each 0,1,..., mq = ) is continuous w.r.t.y and u (from lemma 3.2 in ref. [5] ,...,1} jN =− withUR ⊂ , then the above relations are equivalent to the following minimum principle in blockwise form (for each 0,1,...,1 jN =− , and 1,2,..., iM =

F
pdfFactory Pro trial version www.pdffactory.comEng.& Tech.Journal, Vol.28, No.term in the L.H.S. (left hand side) of the obtained equation by another way, i.e. this equality in the L.H.S. of (20), then summing both sides of the obtained equation, for 0 j = , to 1 jl =−, using the assumptions on the operator (.,.) PDF created with pdfFactory Pro trial version www.pdffactory.comEng.& Tech.Journal, Vol.28, No.and the 4 th terms in the L.H.S. of (23) become positive, and on the other are bounded from the projection theorem and the assumptions on f , then using the discrete Gronwall 's inequality[5], we get that

(
Bu using the discrete integration by part formula on the 1 st term in the L.H.S. of this equation, it becomes 00 From the above convergences, and the assumptions on the functionf , we can passage to the limit in ((,)((,,),) tt yvayvftyuv += , vV ∈ , a.e. in I .i.e. u is a solution of the state equation.

((
twice the 1 st term in L.H.S. of (30), we get 00 in the 1 st term in the L.H.S. of the obtained equation, then subtracting the last obtained equation from (32), we get 0 yy = i.e. the limit point y is a solution to the weak form state equations in the continuous control problem.

1 (
. the limit u of a subsequence of the sequence {} n u satisfies the necessary conditions for optimality for the continuous optimal control problem.Let {} n u be a sequence of admissible classical controls and satisfies the minimum principle in blockwise form for optimality for the discrete control problem.Then the limit of any strongly convergent subsequence of {} n u is admissible and satisfies the pointwise minimum principle form for optimality for the continuous classical optimal control problem.Proof:-Let {} n u be a subsequence (of {} n u , same notation) of admissible classical controls and satisfies the minimum principle in blockwise form (19), for optimality (which studied in[5], theorem 4.2) , i.e. for each pdfFactory Pro trial version www.pdffactory.com

28, No.14, 2010 Convergence of The Discrete Classical Optimal Control Problem To The Continuous Classical Optimal Control Problem Including A Nonlinear Hyperbolic P.D. Equation
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Eng. & Tech. Journal, Vol.28, No.14, 2010 Convergence of The Discrete Classical Optimal Control Problem To The Continuous Classical Optimal Control Problem Including A Nonlinear Hyperbolic P.D. Equation
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28, No.14, 2010 Convergence of The Discrete Classical Optimal Control Problem To The Continuous Classical Optimal Control Problem Including A Nonlinear Hyperbolic P.D. Equation
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& Tech. Journal, Vol.28, No.14, 2010 Convergence of The Discrete Classical Optimal Control Problem To The Continuous Classical Optimal Control Problem Including A Nonlinear Hyperbolic P.D. Equation
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Journal, Vol.28, No.14, 2010 Convergence of The Discrete Classical Optimal Control Problem To The Continuous Classical Optimal Control Problem Including A Nonlinear Hyperbolic P.D. Equation
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Journal, Vol.28, No.14, 2010 Convergence of The Discrete Classical Optimal Control Problem To The Continuous Classical Optimal Control Problem Including A Nonlinear Hyperbolic P.D. Equation
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