Surveys in Mathematics and its Applications
ISSN 1842-6298 (electronic), 1843-7265 (print)
Volume 4 (2009), 139 -- 153
COMPUTING OPTIMAL CONTROL WITH A QUASILINEAR PARABOLIC PARTIAL DIFFERENTIAL EQUATION
M. H. Farag
Abstract. This paper presents the numerical solution of a constrained optimal control problem (COCP) for quasilinear parabolic equations. The COCP is converted to unconstrained optimization problem (UOCP) by applying the exterior penalty function method. Necessary optimality conditions for the considered problem are established. The computing optimal controls are helped to identify the unknown coefficients of the quasilinear parabolic equation. Numerical results are reported.2000 Mathematics Subject Classification: 49J20; 49K20; 49M29; 49M30.
Keywords: Optimal control; Parabolic Equation; Penalty function methods; Existence theory; Necessary optimality conditions.
A. Belmiloudi, On some control problems for nonlinear parabolic equations, Inter. J. of Pure and Appl. Math., Vol. 11, (2) (2004), 115-169. MR2193005(2006g:49066). Zbl 1140.49028.
E. Casas, J.-P. Raymond and H. Zidani, Pontryagin's principle for local solutions of control problems with mixed control-state constraints, SIAM J. on Control and Optim., Vol. 39, (4) (2000), 1182-1203. MR1814272(2001m:49034). Zbl 0894.49011.
N. Damaen, New algorithm for extreme temperature measurements,Adv. In Eng. Soft. 31 (2000), 275-285. Zbl 0962.74042.
M. H. Farag, On the derivation of discrete conjugate boundary value problem for an optimal control parabolic problem, New Zealand Journal of Mathematics, 32 (2003), 21-31. MR1982998(2004b:49047). Zbl 1041.49023.
M. H. Farag, Necessary optimality conditions for constrained optimal control problems governed by parabolic equations, Journal of Vibration and Control ,9 (2003), 949-963. MR1992766(2004f:49048). Zbl 1047.49023.
M. H. Farag, A stability theorem for a class of distributed parameter control systems, Rocky Mountain Journal of Mathematices,Vol. 36 3 (2006), 931-947. MR2254370(20072e:49053). Zbl 1152.49033.
M. Goebel, On existence of optimal control, Math. Nuchr.,93 (1979), 67-73. MR0579843(82f:49005). Zbl 0435.49006.
A. D. Iskenderov and R.K Tagiev, Optimization problems with controls in coefficients of parabolic equations, Differentsial'nye Uravneniya, Vol. 19 (8) (1983), 1324-1334. MR0635177(82m:49005). Zbl 0521.49016.
W. Krabs, Optimization and approximations,Wiley, New York 1979. MR0538272(80j:41003).
O. A. Ladyzhenskaya, Boundary value problems of mathematical physics, Nauka, Moscow, Russian, 1973. MR0599579(58 #29032).
O. A. Ladyzhenskaya, V. A. Solonnikov and N. N. Ural'tseva, Linear and quasilinear parabolic equations, Nauka, Moscow, Russian, 1976. Zbl 0164.12302.
J.-L Lions, Optimal control by systems described by partial differential equations, Mir, Moscow, Russian, 1972. MR0271512(42 #6395). Zbl 0203.09001.
V. P. Mikhailov, Partial differential equations, Nauka, Moscow, Russian, 1983. MR0701394(84f:35002).
A. N. Tikhonov and N. Ya. Arsenin, Methods for the solution of incorrectly posed problems, Nauka, Moscow, Russian, 1974. MR0455366(56 #13605a). Zbl 0499.65030.
F. Troltzsch and A. Unger, Fast solution of optimal control problems in the selective cooling of steel, ZAMM, 81 (2001), 447-456. MR1844451. Zbl 0993.43024.
F. Vassiliev, Numerical Methods for solving extremal problems, Nauka, Moscow, Russian, 1980. MR1002325(90g:65005).
A-Q. Xing, The exact penalty function method in constrained optimal control problems, J. Math. Anl. and Applics., 186 (1994), 514-522. MR1293008(95d:49049). Zbl 0817.49031.
J. Yin and W. Huang, Optimal boundary control of a nonlinear diffusion equation, Apll. Math. E-Notes, 1(2001), 97-103. MR1799683(2004d:49007). Zbl 1010.49002.
M. H. Farag
Department of Mathematics,
Faculty of Science,