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Optimal control of quasilinear parabolic equations*

Published online by Cambridge University Press:  14 November 2011

Eduardo Casas ,
Luis A. Fernández  and
Jiongmin Yong
Eduardo Casas
Affiliation:
Departamento de Matemática Aplicada y Ciencias de la Computación, E.T.S.I. de Caminos, Universidad de Cantabria, 39071 Santander, Spain
Luis A. Fernández
Affiliation:
Departamento de Matemáticas, Estadística y Computación, Facultad de Ciencias, Universidad de Cantabria, 39071 Santander, Spain
Jiongmin Yong
Affiliation:
Department of Mathematics, Fudan University, Shanghai 200433, China
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Extract

This paper deals with optimal control problems governed by quasilinear parabolic equations in divergence form, whose cost functional is of Lagrangian type. Our aim is to prove the existence of solutions and derive some optimality conditions. To attain this second objective, we accomplish the sensitivity analysis of the state equation with respect to the control, proving that, under some assumptions, this relation is Gâteaux differentiable. Finally, a regularising procedure along with Ekeland's variational principle allow us to treat some other problems for which this differentiability property cannot be stated.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 125 , Issue 3 , 1995 , pp. 545 - 565
Copyright
Copyright © Royal Society of Edinburgh 1995

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References

1Ahmed, N. U.. Optimal control of a class of strongly nonlinear parabolic systems. J. Math. Anal. Appl. 61 (1977), 188–207.CrossRefGoogle Scholar
2Barbu, V.. Optimal control of variational inequalities (Boston: Pitman, 1984).Google Scholar
3Casas, E. and Fernández, L. A.. Optimal control of quasilinear elliptic equations with non differentiable coefficients at the origin. Rev. Mat. Univ. Complut. Madrid 4 (1991), 227–50.Google Scholar
4Casas, E. and Fernández, L. A.. Distributed control of systems governed by a general class of quasilinear elliptic equations. J. Differential Equations 104 (1993), 20–47.CrossRefGoogle Scholar
5Casas, E. and Yong, J.. Maximum principle for state-constrained optimal control problems governed by quasilinear elliptic equations (Technical Report 1063, IMA, November 1992).Google Scholar
6Chryssoverghi, I.. Nonconvex optimal control of nonlinear monotone parabolic systems. Systems Control Lett. 8 (1986), 55–62.CrossRefGoogle Scholar
7Ekeland, I.. Nonconvex minimization problems. Bull. Amer. Math. Soc. 1 (1979), 76–91.CrossRefGoogle Scholar
8Ekeland, I. and Temam, R.. Analyse Convexe et Problèmes Variationnels (Paris: Dunod-Gauthier Villars, 1974).Google Scholar
9Fattorini, H. O.. Optimal control problems for distributed parameter systems governed by semilinear parabolic equations in L 1 and L ∞ spaces. In Optimal Control of Partial Differential Equations, Hoffmann, K. H. and Krabs, W., eds, Lecture Notes in Control and Information Sciences 149, pp. 68–80 (Berlin: Springer, 1991).CrossRefGoogle Scholar
10Ladyzhenskaya, O. A., Solonnikov, V. A. and Ural'tseva, N. N.. Linear and Quasilinear Equations of Parabolic Type (Providence, R.I.: American Mathematical Society, 1968).CrossRefGoogle Scholar
11Lions, J. L.. Optimisation pour certaines classes d'équations d'évolution non linéaires. Ann. Mat. Pura Appl. 72 (1966), 275–93.CrossRefGoogle Scholar
12Lions, J. L.. Quelques Méthodes de Résolution des Problèmes aux Limites non Linéaires (Paris: Dunod, 1969).Google Scholar
13Nabadan, S. and Noussair, E. S.. Existence theorems for optimal control of quasilinear parabolic partial differential equations. J. Austral. Math. Soc. 21 (1979), 90–101.CrossRefGoogle Scholar
14Noussair, E. S., Nabadan, S. and Teo, K. L.. On the existence of optimal controls for quasilinear parabolic partial differential equations. J. Optim. Theory Appl. 34 (1981), 99–115.CrossRefGoogle Scholar
15Pan, L. and Yong, J. Optimal control for quasilinear retarded parabolic systems. Math. Syst. Estim. Control (to appear).Google Scholar
16Papageorgiou, N. S.. On the optimal control of strongly nonlinear evolution equations. J. Math. Anal. Appl. 164 (1992), 83–103.CrossRefGoogle Scholar
17Tiba, D.. Optimality conditions for distributed control problems with nonlinear state equations. SIAM J. Control Optim. 23 (1985), 85–110.CrossRefGoogle Scholar
18Tiba, D.. Optimal control of nonsmooth distributed parameter systems, Lecture Notes in Mathematics 1459 (Berlin: Springer, 1990).CrossRefGoogle Scholar
19Tröltzsch, F.. Optimality Conditions for Parabolic Control Problems and Applications (Leipzig: Teubner-Texte, 1984).Google Scholar