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Abstrakty
A problem of sparse optimal boundary control for a semilinear parabolic partial differential equation is considered, where pointwise bounds on the control and mixed pointwise control-state constraints are given. A standard quadratic objective functional is to be minimized that includes a Tikhonov regularization term and the L1-norm of the control accounting for the sparsity. Applying a recent linearization theorem, we derive first-order necessary optimality conditions in terms of a variational inequality under linearized mixed control state constraints. Based on this preliminary result, a Lagrange multiplier rule with bounded and measurable multipliers is derived and sparsity results on the optimal control are demonstrated.
Czasopismo
Rocznik
Tom
Strony
89--124
Opis fizyczny
Bibliogr. 19 poz., rys.
Twórcy
autor
- Departamento de Matem´atica Aplicada y Ciencias de la Computaci´on, E.T.S.I. Industriales y de Telecomunicacion, Universidad de Cantabria, 39005 Santander, Spain
autor
- Institut fur Mathematik, Technische Universitat Berlin, D-10623 Berlin, Germany
Bibliografia
- [1] Bonnans, F. and Shapiro, A. (2000), Perturbation Analysis of Optimization Problems. Springer-Verlag, New York.
- [2] Casas, E., Herzog, R. and Wachsmuth, G. (2012) Optimality conditions and error analysis of semilinear elliptic control problems with L1 cost functional. SIAM J. Optim. 22 (3), 795–820.
- [3] Casas, E. and Tr¨oltzsch, F. (2018a) Sparse optimal control for the heat equation with mixed control-state constraints. To appear in Math. Control Rel. Fields.
- [4] Casas, E. and Tr¨oltzsch, F. (2018b) State-constrained semilinear elliptic optimization problems with unrestricted sparse controls. To appear in Math. Control Rel. Fields.
- [5] Collatz, L. (1952) Aufgaben monotoner Art. Arch. Math., 3, 366–376.
- [6] Collatz, L. (1964) Funktionalanalysis und numerische Mathematik. Die Grundlehren der mathematischen Wissenschaften, 120, Springer-Verlag, Berlin.
- [7] Dautray, R. and Lions, J. (2000) Mathematical Analysis and Numerical Methods for Science and Technology, 5. Springer-Verlag, BerlinHeidelberg-New York.
- [8] Glashoff, K. and Werner, B. (1979) Inverse monotonicity of monotone Loperators with applications to quasilinear and free boundary value problems. J. Math. Anal. Appl., 72 (1), 89–105.
- [9] Grinold, R.C. (1970) Symmetric duality for continuous linear programs. SIAM J. Appl. Math., 18, 84–97.
- [10] Grisvard, P. (1985) Elliptic Problems in Nonsmooth Domains. Pitman, Boston-London-Melbourne. Krabs, W. (1968) Lineare Optimierung in halbgeordneten Vektorr¨aumen. Numer. Math., 11, 220–231.
- [11] Ladyzhenskaya, O., Solonnikov, V., and Ural’tseva, N. (1988) Linear and Quasilinear Equations of Parabolic Type. American Mathematical Society, Providence.
- [12] Ortega, J.M. and Rheinboldt, W.C. (1967) Monotone iterations for nonlinear equations with application to Gauss-Seidel methods. SIAM J. Numer. Anal., 4, 171–190.
- [13] Pao, C.V. (1992) Nonlinear Parabolic and Elliptic Equations. Plenum Press, New York.
- [14] Raymond, J.P. and Zidani, H. (1998) Pontryagin’s principle for stateconstrained control problems governed by parabolic equations with unbounded controls. SIAM J. Control Optim., 36 (6), 1853–1879.
- [15] Raymond, J.R. and Zidani, H. (1999) Hamiltonian Pontryagin’s principles for control problems governed by semilinear parabolic equations. Appl. Math. Optim., 39 (2), 143–177.
- [16] Redheffer, R. and Walter, W. (1979) Inequalities involving derivatives. Pacific J. Math., 85 (1), 165–178.
- [17] Rheinboldt, W.C. (1969) On M-functions and their application to nonlinear Gauss-Seidel iterations and to network flows. BMwF-GMD-23, Gesellschaft fu¨r Mathematik und Datenverarbeitung, Bonn.
- [18] Rosch, A. and Tr¨oltzsch, F. (2007) On regularity of solutions and Lagrange multipliers of optimal control problems for semilinear equations with mixed pointwise control-state constraints. SIAM J. Control and Optimization, 46 (3), 1098–1115.
- [19] Temam, R. (1979) Navier-Stokes Equations. North-Holland, Amsterdam. Tr¨oltzsch, F. (1979) A minimum principle and a generalized bang-bang principle for a distributed optimal control problem with constraints on control and state. Z. Angew. Math. Mech., 59 (12), 737–739.
Uwagi
PL
Opracowanie rekordu ze środków MNiSW, umowa Nr 461252 w ramach programu "Społeczna odpowiedzialność nauki" - moduł: Popularyzacja nauki i promocja sportu (2020)
Typ dokumentu
Bibliografia
Identyfikator YADDA
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