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# THE GRADIENT PROJECTION METHOD FOR SOLVING AN OPTIMAL CONTROL PROBLEM

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M. H. F A R A G (Minia)

l

\

0

2

T

\

0

1

2

t

2

xx

2

x

x

1

2

1

2

### (x, t) the density of heat sources, and the control v is in

1991 Mathematics Subject Classification: 49J20, 49M07, 65L10, 65K10.

Key words and phrases: optimal control, gradient methods, boundary value problems, distributed parameter systems.

[141]

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(3)

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(5)

v

1

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1

21

2

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l

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k

k1

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1k+1

1k

k

v

1k

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1

k1

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2k

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v

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2

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Tl 0

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γ≥0

γ≥0

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k

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4

### The proof directly follows from that of Theorem 5.2.1 of [19].

References

[1] A. G. B u t k o v s k i˘ı, Optimal Control Theory for Systems with Distributed Param- eters, Nauka, Moscow, 1965 (in Russian).

[2] Yn. V. E g o r o v, On some optimal control problems, Zh. Vychisl. Mat. i Mat. Fiz.

3 (1963), 887–904 (in Russian).

[3] M. H. F a r a g, A numerical solution to a nonlinear problem of the identification of the characteristics of a mathematical model of heat exchange, in: Mathematical Modeling and Automated Systems, A. D. Iskenderov (ed.), Bakin. Gos. Univ., Baku, 1990, 23–30 (in Russian).

[4] M. H. F a r a g and S. H. F a r a g, An existence and uniqueness theorem for one optimal control problem, Period. Math. Hungar. 30 (1995), 61–65.

[5] A. F r i e d m a n, Partial Differential Equations of Parabolic Type, Prentice-Hall, En- glewood Cliffs, N.J., 1964.

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[6] A. D. I s k e n d e r o v, On a certain inverse problem for quasilinear parabolic equa- tions, Differentsial’nye Uravneniya 10 (1974), 890–898 (in Russian).

[7] A. D. I s k e n d e r o v and R. K. T a g i e v, Optimization problems with controls in coefficients of parabolic equations, ibid. 19 (1983), 1324–1334 (in Russian).

[8] J.-L. L i o n s, Control problems in systems described by partial differential equations, in: Mathematical Theory of Control , A. V. Balakrishnan and L. W. Neustadt (eds.), Academic Press, New York and London, 1969, 251–271.

[9] —, Optimal Control by Systems Described by Partial Differential Equations, Mir, Moscow, 1972 (in Russian).

[10] K. A. L u r i e, Optimal Control in Problems of Mathematical Physics, Nauka, Moscow, 1975 (in Russian).

[11] M. D. M a d a t o v, Regularization of one class of optimal control problems, in: Ap- proximate Methods and Computer, A. D. Iskenderov (ed.), Bakin. Gos. Univ., Baku, 1982, 78–80 (in Russian).

[12] A. M o k r a n e, An existence result via penalty method for some nonlinear parabolic unilateral problems, Boll. Un. Mat. Ital. B 8 (1994), 405–417.

[13] G. A. P h i l l i p s o n and S. K. M i t t e r, Numerical solution of a distributed iden- tification problem via a direct method, in: Computing Methods in Optimization Problems—2, L. A. Zadeh, L. W. Neustadt and A. V. Balakrishnan (eds.), Aca- demic Press, New York, 1969, 305–315.

[14] E. P o l a k, Computational Methods in Optimization, Academic Press, New York, 1971.

[15] B. N. P s h e n i c h n y˘ı and Yu. M. D a n i l i n, Numerical Methods in Extremal Prob- lems, Mir, Moscow, 1982.

[16] J. B. R o s e n, The gradient projection method for nonlinear programming. Part I : Linear constraints, SIAM J. Appl. Math. 8 (1960), 181–217.

[17] —, The gradient projection method for nonlinear programming. Part II : Nonlinear constraints, ibid. 9 (1961), 514–532.

[18] Ts. T s a c h e v, Optimal control of linear parabolic equation: The constrained right- hand side as control function, Numer. Funct. Anal. Optim. 13 (1992), 369–380.

[19] F. P. V a s i l’ e v, Numerical Methods for Solving Extremal Problems, Nauka, Moscow, 1988 (in Russian).

M. H. Farag

Department of Mathematics Faculty of Science

Minia University Minia, Egypt

revised version on 16.2.1996

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