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Second-order optimality conditions for the bilinear optimal control of a degenerate equation. (English) Zbl 07916288

MSC:

35Q93 PDEs in connection with control and optimization
35B50 Maximum principles in context of PDEs
35K65 Degenerate parabolic equations
49J20 Existence theories for optimal control problems involving partial differential equations
49K20 Optimality conditions for problems involving partial differential equations
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[1] Epstein, CL, Mazzeo, R.Degenerate diffusion operators arising in population biology. Princeton: Princeton University Press; 2013. · Zbl 1309.47001
[2] Ghil, M.Climate stability for a sellers-type model. J Atmos Sci. 1976;33(1):3-20. doi:
[3] North, GR, Kim, K-Y.Energy balance climate models. Weinheim: John Wiley & Sons; 2017.
[4] Budyko, MI.The effect of solar radiation variations on the climate of the earth. tellus. 1969;21(5):611-619. doi:
[5] Sellers, WD. A global climatic model based on the energy balance of the earth-atmosphere system. Journal of Applied Meteorology (1962-1982). 1969. p. 392-400.
[6] Aronna, MS. Singular solutions in optimal control: second order conditions and a shooting algorithm. arXiv preprint arXiv:1210.7425, 2012.
[7] Aronna, MS, Bonnans, JF, Goh, BS.Second order analysis of control-affine problems with scalar state constraint. Math Program. 2016;160(1-2):115-147. doi: · Zbl 1352.49020
[8] Aronna, MS, Bonnans, JF, Kroner, A.Optimal control of pdes in a complex space setting: application to the Schrödinger equation. SIAM J Control Optim. 2019;57(2):1390-1412. doi: · Zbl 1412.49051
[9] Aronna, MS, Bonnans, JF, Kröner, A.State-constrained control-affine parabolic problems i: first and second order necessary optimality conditions. Set-Valued Var Anal. 2021;29(2):383-408. doi: · Zbl 1470.49007
[10] Aronna, MS, Bonnans, JF, Kröner, A.State constrained control-affine parabolic problems ii: second order sufficient optimality conditions. SIAM J Control Optim. 2021;59(2):1628-1655. doi: · Zbl 1465.49003
[11] Aronna, MS, Bonnans, JF, Kröner, A.Optimal control of infinite dimensional bilinear systems: application to the heat and wave equations. Math Program. 2018;168(1-2):717-757. doi: · Zbl 1454.49028
[12] Aronna, MS, Tröltzsch, F.First and second order optimality conditions for the control of Fokker-Planck equations. ESAIM Control Optim Calc Var. 2021;27:15. doi: · Zbl 1467.49002
[13] Tory, EM, Karlsen, KH, Bürger, R, et al. Strongly degenerate parabolic-hyperbolic systems modeling polydisperse sedimentation with compression. SIAM J Appl Math. 2003;64(1):41-80. doi: · Zbl 1047.35071
[14] Shimakura, N.Partial differential operators of elliptic type. Providence: Amer Mathematical Society; 1992. · Zbl 0757.35015
[15] Casas, E, Tröltzsch, F.Second order analysis for optimal control problems: improving results expected from abstract theory. SIAM J Optim. 2012;22(1):261-279. doi: · Zbl 1259.90162
[16] Diaz, J, Hetzer, G, Tello, L.An energy balance climate model with hysteresis. Nonlinear Anal Theory Methods Appl. 2006;64(9):2053-2074. doi: · Zbl 1098.35090
[17] Díaz, JID, del Castillo, LT. A nonlinear parabolic problem on a Riemannian manifold without boundary arising in climatology. Collectanea Mathematica, 1999. p. 19-51. · Zbl 0936.35095
[18] North, GR.Analytical solution to a simple climate model with diffusive heat transport. J Atmos Sci. 1975;32(7):1301-1307. doi:
[19] Cannarsa, P, Floridia, G.Approximate multiplicative controllability for degenerate parabolic problems with robin boundary conditions. Commun Appl Ind Math. 2011;2(2):1-16. · Zbl 1329.93029
[20] Floridia, G.Approximate controllability for nonlinear degenerate parabolic problems with bilinear control. J Differ Equ. 2014;257(9):3382-3422. doi: · Zbl 1294.93022
[21] Floridia, G. Nonnegative controllability for a class of nonlinear degenerate parabolic equations with application to climate science. arXiv preprint arXiv:2003.04966, 2020.
[22] Cannarsa, P, Floridia, G. Approximate controllability for linear degenerate parabolic problems with bilinear control. arXiv preprint arXiv:1106.4232, 2011.
[23] Arab, Z, Tunç, C.Well-posedness and regularity of some stochastic time-fractional integral equations in Hilbert space. J Taibah Univ Sci. 2022;16(1):788-798. doi:
[24] Salim, A, Mesri, F, Benchohra, M, et al. Controllability of second order semilinear random differential equations in fréchet spaces. Mediterr J Math. 2023;20(2):84. doi: · Zbl 1505.34098
[25] Tunç, C, Tunç, O, Wen, C-F, et al. On the qualitative analyses solutions of new mathematical models of integro-differential equations with infinite delay. Mathematical Methods in the Applied Sciences, 2023.
[26] Lions, JL.Equations différentielles opérationnelles: et problèmes aux limites. Berlin: Springer-Verlag; 2013.
[27] Dorville, R. Bilinear boundary optimal control with final observation for the heat equation. Applicable Analysis, 2021.
[28] Kenne, C, Leugering, G, Mophou, G.Optimal control of a population dynamics model with missing birth rate. SIAM J Control Optim. 2020;58:1289-1313. doi: · Zbl 1453.49004
[29] Kenne, C, Nkemzi, B. Optimal control of averaged state of a population dynamics model. In: Studies in Evolution Equations and Related Topics. Springer; 2021. p. 113-127. · Zbl 1481.49006
[30] Kenne, C, Zongo, P, Dorville, R, et al. Optimal control of a coupled degenerate population dynamics model with unknown birth rates. Nonlinear Stud. 2021;28(4):1225-1252.
[31] Mophou, G, Kéré, M, Njoukoué, LLD.Robust hierarchic control for a population dynamics model with missing birth rate. Math Control Signals Syst. 2020;32(2):209-239. doi: · Zbl 1448.92239
[32] Kenne, C. Sur les modèles de dynamique de populations et l’émergence de la maladie dans les eaux douces [PhD thesis]. Université des Antilles; 2022.
[33] Casas, E, Tröltzsch, F.Second order optimality conditions and their role in pde control. Jahresber Dtsch Math Ver. 2015;117(1):3-44. doi: · Zbl 1311.49002
[34] Tröltzsch, F.Optimal control of partial differential equations: theory, methods, and applications. Providence: American Mathematical Soc.; 2010. · Zbl 1195.49001
[35] Kenne, C, Mophou, G, Warma, M. Bilinear optimal control for a fractional diffusive equation. arXiv preprint arXiv:2210.17494, 2022.
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