Abstract
In the second order analysis of infinite dimension optimization problems, we have to deal with the so-called two-norm discrepancy. As a consequence of this fact, the second order optimality conditions usually imply local optimality in the L ∞ sense. However, we have observed that the L 2 local optimality can be proved for many control problems of partial differential equations. This can be deduced from the standard second order conditions. To this end, we make some quite realistic assumptions on the second derivative of the cost functional. These assumptions do not hold if the control does not appear explicitly in the cost functional. In this case, the optimal control is usually of bang-bang type. For this type of problems we also formulate some new second order optimality conditions that lead to the strict L 2 local optimality of the bang-bang controls.
This work was partially supported by the Spanish Ministerio de Economía y Competitividad under project MTM2011-22711.
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Bonnans, F., Shapiro, A.: Perturbation Analysis of Optimization Problems. Springer (2000)
Bonnans, J.: Second-order analysis for control constrained optimal control problems of semilinear elliptic systems. Appl. Math. Optim. 38, 303–325 (1998)
Casas, E.: Boundary control of semilinear elliptic equations with pointwise state constraints. SIAM J. Control Optim. 31, 993–1006 (1993)
Casas, E.: Second order analysis for bang-bang control problems of pde (submitted, 2012)
Casas, E., Mateos, M.: Second order optimality conditions for semilinear elliptic control problems with finitely many state constraints. SIAM J. Control Optim. 40, 1431–1454 (2002)
Casas, E., Tröltzsch, F.: Optimality conditions for a class of optimal control problems with quasilinear elliptic equations. SIAM J. Control Optim. 48, 688–718 (2009)
Casas, E., Tröltzsch, F.: A general theorem on error estimates with application to elliptic optimal control problems. Comp. Optim. Appls. (to appear)
Casas, E., Tröltzsch, F.: Second order analysis for optimal control problems: Improving results expected from abstract theory. SIAM J. Optim. (to appear)
Dunn, J.: Second-order optimality conditions in sets of L ∞ functions with range in a polyhedron. SIAM J. Control Optim. 33, 1603–1635 (1995)
Gilbarg, D., Trudinger, N.: Elliptic Partial Differential Equations of Second Order. Springer, Heidelberg (1977)
Haller-Dintelmann, R., Meyer, C., Rehberg, J., Schiela, A.: Hölder continuity and optimal control for nonsmooth elliptic domains. Appl. Math. Optim. 60, 397–428 (2009)
Stampacchia, G.: Problemi al contorno ellittici con dati discontinui dotati di soluzioni Hölderiane. Ann. Mat. Pura Appl. 51, 1–38 (1960)
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Casas, E. (2013). Second Order Conditions for L 2 Local Optimality in PDE Control. In: Hömberg, D., Tröltzsch, F. (eds) System Modeling and Optimization. CSMO 2011. IFIP Advances in Information and Communication Technology, vol 391. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-36062-6_1
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DOI: https://doi.org/10.1007/978-3-642-36062-6_1
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