Controllability of a class of infinite dimensional systems with age structure

• Autorzy: Maity Debayan , Tucsnak Marius , Zuazua Enrique
• Języki publikacji: EN

Abstrakty

EN

Given a linear dynamical system, we investigate the linear infinite dimensional system obtained by grafting an age structure. Such systems appear essentially in population dynamics with age structure when phenomena like spatial diffusion or transport are also taken into consideration. We first show that the new system preserves some of the wellposedness properties of the initial one. Our main result asserts that if the initial system is null controllable in a time small enough then the structured system is also null controllable in a time depending on the various involved parameters.

Słowa kluczowe

EN

infinite dimensional linear system; age structure; admissible control operator; null controllability; population dynamics

• Wydawca: Systems Research Institute, Polish Academy of Sciences
• Czasopismo: Control and Cybernetics
• Rocznik: 2019
• Tom: Vol. 48, No. 2
• Strony: 231--260
• Opis fizyczny: Bibliogr. 32 poz., rys.

Twórcy

autor

Maity Debayan

• Chair of Computational Mathematics, Deusto Foundation, 48007 Bilbao, Basque Country, Spain
• Departamento de Matemáticas, Universidad Aut´onoma de Madrid, 28049 Madrid, Spain

autor

Tucsnak Marius

• Institut de Mathématiques, Université de Bordeaux, Bordeaux INP, CNRS, F-33400 Talence, France

autor

Zuazua Enrique

• Chair in Applied Analysis, Alexander von Humboldt-Professorship, Department of Mathematics, Friedrich-Alexander-Universität Erlangen-Nürnberg, 91058 Erlangen, Germany
• Chair of Computational Mathematics, Deusto Foundation, 48007 Bilbao, Basque Country, Spain
• Departamento de Matemáticas, Universidad Aut´onoma de Madrid, 28049 Madrid, Spain

Bibliografia

• Ainseba, B. (2002) Exact and approximate controllability of the age and space population dynamics structured model. J. Math. Anal. Appl., 275(2): 562–574.
• Ainseba, B. and Anita, S. (2001) Local exact controllability of the agedependent population dynamics with diffusion. Abstr. Appl. Anal., 6(6): 357–368.
• Ainseba, B. and Anita, S. (2004) Internal exact controllability of the linear population dynamics with diffusion. Electron. J. Differential Equations, Paper no. 112, 11.
• Ainseba, B., Echarroudi, Y., Maniar, L. et al. (2013) Null controllability of a population dynamics with degenerate diffusion. Differential and Integral Equations, 26(11/12): 1397–1410.
• Anantharaman, N., Léautaud, M. and Maciá, F. (2016) Wigner measures and observability for the Schrödinger equation on the disk. Inventiones Mathematicae, 206(2): 485–599.
• Barbu, V., Iannelli, M. and Martcheva, M. (2001) On the controllability of the Lotka-McKendrick model of population dynamics. J. Math. Anal. Appl., 253(1): 142–165.
• Boutaayamou, I. and Echarroudi, Y. (2017) Null controllability of a population dynamics with interior degeneracy. arXiv preprint arXiv:1704.00936.
• Brikci, F. B., Clairambault, J., Ribba, B. and Perthame, B. (2008) An age-and-cyclin-structured cell population model for healthy and tumoral tissues. Journal of Mathematical Biology, 57(1): 91–110.
• Cannarsa, P., Fragnelli, G. and Rocchetti, D.(2008) Controllability results for a class of one-dimensional degenerate parabolic problems in nondivergence form. J. Evol. Equ., 8(4): 583–616.
• Cannarsa, P., Martinez, P. and Vancostenoble, J. (2009) Carleman estimates and null controllability for boundary-degenerate parabolic operators. Comptes Rendus Mathematique, 347: 147–152.
• Cannarsa, P., Martinez, P. and Vancostenoble, J. (2016) Global Carleman Estimates for Degenerate Parabolic Operators with Applications, 239. American Mathematical Society.
• Curtain, R. F. and Weiss, G. (1989) Well posedness of triples of operators (in the sense of linear systems theory). In: Control and Estimation of Distributed Parameter Systems (Vorau, 1988), Internat. Ser. Numer. Math., 91, Birkhäuser, Basel, 41–59.
• Fragnelli, G. (2018) Carleman estimates and null controllability for a degenerate population model. Journal de Mathématiques Pures et Appliquées, 115: 74–126.
• Fursikov, A. V. and Imanuvilov, O. Y.(1996) Controllability of evolution equations. Lecture Notes Series, 34. Seoul National University, Research Institute of Mathematics, Global Analysis Research Center, Seoul.
• Gurtin, M. E. (1973) A system of equations for age-dependent population diffusion. Journal of Theoretical Biology, 40(2): 389–392.
• Hegoburu, N. and Anita, S. (2019) Null controllability via comparison results for nonlinear age-structured population dynamics. Math. Control Signals Systems, 31(1): Art. 2, 38.
• Hegoburu, N., Magal, P. and Tucsnak, M.(2018) Controllability with positivity constraints of the Lotka-McKendrick system. SIAM Journal on Control and Optimization, 56(2): 723–750.
• Hegoburu, N. and Tucsnak, M. (2018) Null controllability of the Lotka-McKendrick system with spatial difusion. Math. Control Relat. Fields, 8(3-4): 707–720.
• Jaffard, S. (1988) Contróle interne exact des vibrations d’une plaque carrée. C. R. Acad. Sci. Paris S´er. I Math., 307(14): 759–762.
• Magal, P. and Ruan, S. (2018) Theory and applications of abstract semilinear Cauchy problems. Applied Mathematical Sciences, 201: 978–3.
• Maity, D. (2019) On the Null Controllability of the Lotka-McKendrick System. Math. Control Relat. Fields, 9(4): 719–728.
• Maity, D., Tucsnak, M. and Zuazua, E. (2019) Controllability and positivity constraints in population dynamics with age structuring and diffusion. Journal de Math´ematiques Pures et Appliqu´ees, 129: 153–179.
• Micu, S., Roventa, I. and Tucsnak, M. (2012) Time optimal boundary controls for the heat equation. Journal of Functional Analysis, 263(1): 25–49.
• Micu, S. and Zuazua, E. (2006) On the controllability of a fractional order parabolic equation. SIAM J. Control Optim., 44(6): 1950–1972.
• Miller, L. (2006) On the controllability of anomalous difusions generated by the fractional Laplacian. Math. Control Signals Systems, 18(3): 260–271.
• Pighin, D. and Zuazua, E. (2018) Controllability under positivity constraints of multi-d wave equations. arXiv preprint arXiv:1804.02151.
• Seidman, T. I. (1976) Observation and prediction for the heat equation. III. J. Diferential Equations, 20(1): 18–27.
• Tenenbaum, G. and Tucsnak, M. (2011) On the null-controllability of difusion equations. ESAIM Control Optim. Calc. Var., 17(4): 1088–1100.
• Tucsnak, M. and Weiss, G.(2009) Observation and Control for Operator Semigroups. Birkhäuser Advanced Texts: Basler Lehrbücher. [Birkhäuser Advanced Texts: Basel Textbooks]. Birkhäuser Verlag, Basel.
• Walker, C. (2013) Some remarks on the asymptotic behavior of the semigroup associated with age-structured difusive populations. Monatsh. Math., 170(3-4): 481–501.
• Webb, G. F. (1985) Theory of nonlinear age-dependent population dynamics. Monographs and Textbooks in Pure and Applied Mathematics, 89, Marcel Dekker, Inc., New York.
• Webb, G. F. (2008) Population models structured by age, size, and spatial position. In: Structured Population Models in Biology and Epidemiology. Lecture Notes in Math. 1936, Springer, Berlin, 1–49.

Uwagi

Opracowanie rekordu ze środków MNiSW, umowa Nr 461252 w ramach programu "Społeczna odpowiedzialność nauki" - moduł: Popularyzacja nauki i promocja sportu (2021).