Abstract
Pontryagin’s principle, originally devised for the optimal control of ordinary differential equation, has recently been extended to the optimal control of semilinear elliptic systems and variational inequalities in the case of a distributed control. In this paper we show that if the control is also active at the boundary of the domain, then a boundary Hamiltonian satisfying a boundary maximum Pontryagin’s principle appears in a natural way.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
J.F. Bonnans. Pontryagin’s principle for the optimal control of semilinear elliptic systems with state constraints. In 30th IEEE Conference on Control and Decision, volume 2, pages 1976–1979, Brighton, England, 1991.
J.F. Bonnans and E. Casas. Contrôle de systèmes elliptiques semilinéaires comportant des contraintes sur l’état. In H. Brezis and J.L. Lions, editors, Nonlinear Partial Differential Equations and Their Applications. College de France Seminar, volume 8, pages 69–86. Longman Scientific & Technical, New York, 1988.
J.F. Bonnans and E. Casas. Optimal control of semilinear multistate systems with state constraints. SIAM J. on Control & Optim., 27 (2): 446–455, 1989.
J.F. Bonnans and E. Casas. Un principe de Pontryagine pour le contrôle des systèmes elliptiques. J. Differential Equations, 90(2): 288–303, 1991.
J.F. Bonnans and E. Casas. An extension of Pontryagin’s principle for state-constrained optimal control of semilinear elliptic equation and variational inequalities. To appear.
J.F. Bonnans and D. Tiba. Pontryagin’s principle in the control of semilinear elliptic variational equations. J. Appl. Math. and Optim., 23: 299–312, 1991.
J.V. Burke. Calmness and exact penalization. SIAM J. on Control & Optim., 29(2): 493–497, 1991.
E. Casas. Control of an elliptic problem with pointwise state constraints. SIAM J. on Control & Optim., 24(6): 1309–1318, 1986.
E. Casas. Boundary control of semilinear elliptic equations with pointwise state constraints. SIAM J. on Control and Optim., to appear.
I. Ekeland. Nonconvex minimization problems. Bull. Amer. Math. Soc, 1(3): 76–91, 1979.
D. Gilbarg and N.S. Trudinger. Elliptic Partial Differential Equations of Second Order. Springer-Verlag, Berlin-Heidelberg-New York, 1977.
U. Mackenroth. On some elliptic optimal control problems with state constraints. Optimization, 17(1986), 595–607.
G. Stampacchia. Le problème de Dirichlet pour les equations elliptiques du second ordre à coefficients discontinus. Ann. Inst. Fourier (Grenoble), 15: 189–258, 1965.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1992 Springer Basel AG
About this chapter
Cite this chapter
Bonnans, F., Casas, E. (1992). A boundary Pontryagin’s principle for the optimal control of state-constrained elliptic systems. In: Barbu, V., Tiba, D., Bonnans, J.F. (eds) Optimization, Optimal Control and Partial Differential Equations. International Series of Numerical Mathematics / Internationale Schriftenreihe zur Numerischen Mathematik / Série Internationale d’Analyse Numérique, vol 107. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-8625-3_22
Download citation
DOI: https://doi.org/10.1007/978-3-0348-8625-3_22
Publisher Name: Birkhäuser, Basel
Print ISBN: 978-3-0348-9704-4
Online ISBN: 978-3-0348-8625-3
eBook Packages: Springer Book Archive