doi: 10.3934/eect.2021053
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$ L^p $-exact controllability of partial differential equations with nonlocal terms

1. 

Department of Sciences and Methods for Engineering, University of Modena and Reggio Emilia, I-42122, Italy

2. 

Department of Physics, Informatics and Mathematics, University of Modena and Reggio Emilia, I-41125, Italy

* Corresponding author: luisa.malaguti@unimore.it

Received  February 2021 Revised  August 2021 Early access October 2021

Fund Project: The authors are members of the Gruppo Nazionale per l'Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM) and acknowledge financial support from this institution

The paper deals with the exact controllability of partial differential equations by linear controls. The discussion takes place in infinite dimensional state spaces since these equations are considered in their abstract formulation as semilinear equations. The linear parts are densely defined and generate strongly continuous semigroups. The nonlinear terms may also include a nonlocal part. The solutions satisfy nonlocal properties, which are possibly nonlinear. The states belong to Banach spaces with a Schauder basis and the results exploit topological methods. The novelty of this investigation is in the use of an approximation solvability method which involves a sequence of controllability problems in finite-dimensional spaces. The exact controllability of nonlocal solutions can be proved, with controls in $ L^p $ spaces, $ 1<p<\infty $. The results apply to the study of the exact controllability for the transport equation in arbitrary Euclidean spaces and for the equation of the nonlinear wave equation.

Citation: Luisa Malaguti, Stefania Perrotta, Valentina Taddei. $ L^p $-exact controllability of partial differential equations with nonlocal terms. Evolution Equations & Control Theory, doi: 10.3934/eect.2021053
References:
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I. BenedettiL. Malaguti and V. Taddei, Nonlocal solutions of parabolic equations with strongly elliptic differential operators, J. Math. Anal. Appl., 473 (2019), 421-443.  doi: 10.1016/j.jmaa.2018.12.059.  Google Scholar

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I. Boutaayamou and G. Fragnelli, A degenerate population system: Carleman estimates and controllability, Nonlinear Anal., 195 (2020), 111742, 29 pp. doi: 10.1016/j.na.2019.111742.  Google Scholar

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K.-J. Engel and R. Nagel, One-Parameter Semigroups for Linear Evolution Equations, Graduate texts in Mathematics, no. 194, Springer-Verlag, New York, 2000.  Google Scholar

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H. R. HenríquezV. Poblete and J. C. Pozo, Mild solutions of non-autonomous second order problems with nonlocal initial conditions, J. Math. Anal. Appl., 412 (2014), 1064-1083.  doi: 10.1016/j.jmaa.2013.10.086.  Google Scholar

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V. Hernández-Santamaría and E. Zuazua, Controllability of shadow reaction-diffusion systems, J. Differential Equations, 268 (2020), 3781-3818.  doi: 10.1016/j.jde.2019.10.012.  Google Scholar

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C. Laurent and L. Rosier, Exact controllability of semilinear heat equations in spaces of analytic functions, Ann. Inst. H. Poincaré Anal. Non Linéaire, 37 (2020), 1047–1073. doi: 10.1016/j.anihpc.2020.03.001.  Google Scholar

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J.-L. Lions, Exact controllability and singular perturbations, Wave motion: Theory, Modelling, and Computation, Math. Sci. Res.Inst. Publ., 7, ch. Exact Controllability and Singular Perturbations. (Berkeley, Calif., 1986), 217–247, Springer, New York, (1987). doi: 10.1007/978-1-4613-9583-6_8.  Google Scholar

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—————, Exact Controllability, Perturbations and Stabilization of Distributed Systems. Vol. 1. With Appendices by E. Zuazua, C. Bardos, G. Lebeau and J. Rauch., Research in Applied Mathematics, no. 8, Masson, Paris, 1988.  Google Scholar

[25]

K. MagnussonA. J. Pritchard and M. D. Quinn, The application of fixed point theorems to global nonlinear controllability problems, Mathematical Control Theory, 14 (1985), 319-344.   Google Scholar

[26]

L. MalagutiS. Perrotta and V. Taddei, Exact controllability of infinite dimensional systems with controls of minimal norm, Topol. Methods Nonlinear Anal., 54 (2019), 1001-1021.  doi: 10.12775/tmna.2019.087.  Google Scholar

[27]

V. Obukhovskii and P. Zecca, Controllability for systems governed by semilinear differential inclusions in a Banach space with a noncompact semigroup, Nonlinear Anal., 70 (2009), 3424-3436.  doi: 10.1016/j.na.2008.05.009.  Google Scholar

[28]

A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Applied Mathematical Sciences, no. 44, Springer, New York, 1983. doi: 10.1007/978-1-4612-5561-1.  Google Scholar

[29]

M. F. Pinaud and H. R. Henríquez, Controllability of systems with a general nonlocal condition, J. Differential Equations, 269 (2020), 4609-4642.  doi: 10.1016/j.jde.2020.03.029.  Google Scholar

[30]

L. S. Pul'kina and A. E. Savenkova, A problem with a nonlocal, with respect to time, condition for multidimensional hyperbolic equations, Russian Math., 60 (2016), 33-43.   Google Scholar

[31]

A. E. Taylor and D. C. Lay, Introduction to Functional Analysis, John Wiley & Sons Inc, New York, 1980.  Google Scholar

[32]

R. Triggiani, A note on the lack of exact controllability for mild solutions in Banach spaces, SIAM J. Control Optim., 15 (1977), 407-411.  doi: 10.1137/0315028.  Google Scholar

[33]

V. Vijayakumar and R. Murugesu, Controllability for a class of second-order evolution differential inclusions without compactness, Appl. Anal., 98 (2019), 1367-1385.  doi: 10.1080/00036811.2017.1422727.  Google Scholar

[34]

I. I. Vrabie, $C_0$ Semigroups and Applications, North-Holland Mathematics Studies, no. 191, North-Holland Publishing Co., Amsterdam, 2003.  Google Scholar

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J. Zabczyk, Mathematical Control Theory. An Introduction. Reprint of the 1995 Edition, Modern Birkhäuser Classics, Birkhäuser Boston Inc., Boston, MA, 2008. doi: 10.1007/978-0-8176-4733-9.  Google Scholar

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E. Zuazua, An Introduction to the Exact Controllability for Distributed Systems, Textos e Notas, C.M.A.F., Universidades de Lisboa, 44 (1990). Google Scholar

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E. Zuazua, Exact controllability for semilinear wave equations in one space dimension, Ann. Inst. H. Poincaré Anal. Non Linéaire, 10 (1993), 109–129. doi: 10.1016/S0294-1449(16)30221-9.  Google Scholar

show all references

References:
[1]

V. Barbu, Controllability and Stabilization of Parabolic Equations, Progress in Nonlinear Differential Equations and their Applications, no. 90, Birkhäuser/Springer, Cham, 2018. doi: 10.1007/978-3-319-76666-9.  Google Scholar

[2]

I. BenedettiL. Malaguti and V. Taddei, Nonlocal solutions of parabolic equations with strongly elliptic differential operators, J. Math. Anal. Appl., 473 (2019), 421-443.  doi: 10.1016/j.jmaa.2018.12.059.  Google Scholar

[3]

I. BenedettiV. Obukhovskii and V. Taddei, Controllability for systems governed by semilinear evolution inclusions without compactness, NoDEA Nonlinear Differential Equations Appl., 21 (2014), 795-812.  doi: 10.1007/s00030-014-0267-0.  Google Scholar

[4]

A. Bensoussan, G. Da Prato, M. C. Delfour and S. K. Mitter, Representation and Control of Infinite-Dimensional Systems, Vol 1, Systems & Control: Foundations & Applications, Birkhäuser Boston, Inc., Boston, MA, 1992. doi: 10.1007/978-1-4612-2750-2.  Google Scholar

[5]

S. Bochner and A. E. Taylor, Linear functionals on certain spaces of abstractly-valued functions, Ann. of Math., 39 (1938), 913-944.  doi: 10.2307/1968472.  Google Scholar

[6]

I. Boutaayamou and G. Fragnelli, A degenerate population system: Carleman estimates and controllability, Nonlinear Anal., 195 (2020), 111742, 29 pp. doi: 10.1016/j.na.2019.111742.  Google Scholar

[7]

H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Universitext, Springer, New York, 2011.  Google Scholar

[8]

L. Byszewski, Theorems about the existence and uniqueness of solutions of a semilinear evolution nonlocal cauchy problem, J. Math. Anal. Appl., 162 (1991), 494-505.  doi: 10.1016/0022-247X(91)90164-U.  Google Scholar

[9]

R. M. ColomboA. Corli and M. D. Rosini, Non local balance laws in traffic models and crystal growth, ZAMM Z. Angew. Math. Mech., 87 (2007), 449-461.  doi: 10.1002/zamm.200710327.  Google Scholar

[10]

C. De Lellis, P. Gwiazda and A. Świerczewska-Gwiazda, Transport equations with integral terms: Existence, uniqueness and stability, Calc. Var. Partial Differential Equations, 55 (2016), Art. 128, 17 pp. doi: 10.1007/s00526-016-1049-9.  Google Scholar

[11]

J. Diestel and J. J. Uhl Jr, Vector Measures, Math. Surveys, no. 15, AMS, Providence, RI, 1977.  Google Scholar

[12]

R. J. DiPerna and P.-L. Lions, Ordinary differential equations, transport theory and Sobolev spaces, Invent. Math., 98 (1989), 511-517.  doi: 10.1007/BF01393835.  Google Scholar

[13]

P. Enflo, A counterexample to the approximation problem in Banach spaces, Acta Math., 130 (1973), 309-317.  doi: 10.1007/BF02392270.  Google Scholar

[14]

K.-J. Engel and R. Nagel, One-Parameter Semigroups for Linear Evolution Equations, Graduate texts in Mathematics, no. 194, Springer-Verlag, New York, 2000.  Google Scholar

[15]

R. Glowinski, J.-L. Lions and J. He, Exact and Approximate Controllability for Distributed Parameter Systems. A Numerical Approach, Encyclopedia of Mathematics and its Applications, no. 117, Cambridge University Press, Cambridge, 2008. Google Scholar

[16]

H. R. HenríquezV. Poblete and J. C. Pozo, Mild solutions of non-autonomous second order problems with nonlocal initial conditions, J. Math. Anal. Appl., 412 (2014), 1064-1083.  doi: 10.1016/j.jmaa.2013.10.086.  Google Scholar

[17]

V. Hernández-Santamaría and E. Zuazua, Controllability of shadow reaction-diffusion systems, J. Differential Equations, 268 (2020), 3781-3818.  doi: 10.1016/j.jde.2019.10.012.  Google Scholar

[18]

W. B. Johnson and J. Lindenstrauss, Basic concepts in the geometry of Banach spaces, Handbook of the Geometry of Banach Spaces, Vol I, North-Holland Publishing Co., Amsterdam, (2001), 1–84. doi: 10.1016/S1874-5849(01)80003-6.  Google Scholar

[19]

M. Kamenskii, V. Obukhovskii and P. Zecca, Condensing Multivalued Maps and Semilinear Differential Inclusions in Banach Spaces, Grundlehren Math. Wiss., W. de Gruyter, Berlin, 2001. doi: 10.1515/9783110870893.  Google Scholar

[20]

I. Lasiecka and R. Triggiani, Exact controllability of semilinear abstract systems with application to waves and plates boundary control problems, Appl. Math. Optim., 23 (1991), 109-154.  doi: 10.1007/BF01442394.  Google Scholar

[21]

C. Laurent and L. Rosier, Exact controllability of semilinear heat equations in spaces of analytic functions, Ann. Inst. H. Poincaré Anal. Non Linéaire, 37 (2020), 1047–1073. doi: 10.1016/j.anihpc.2020.03.001.  Google Scholar

[22]

J. Lindenstrauss and L. Tzafriri, Classical Banach Spaces I: Sequence Spaces, Ergebnisse der Mathematik und ihrer Grenzgebiete, no. 92, Springer-Verlag, Berlin-New York, 1977.  Google Scholar

[23]

J.-L. Lions, Exact controllability and singular perturbations, Wave motion: Theory, Modelling, and Computation, Math. Sci. Res.Inst. Publ., 7, ch. Exact Controllability and Singular Perturbations. (Berkeley, Calif., 1986), 217–247, Springer, New York, (1987). doi: 10.1007/978-1-4613-9583-6_8.  Google Scholar

[24]

—————, Exact Controllability, Perturbations and Stabilization of Distributed Systems. Vol. 1. With Appendices by E. Zuazua, C. Bardos, G. Lebeau and J. Rauch., Research in Applied Mathematics, no. 8, Masson, Paris, 1988.  Google Scholar

[25]

K. MagnussonA. J. Pritchard and M. D. Quinn, The application of fixed point theorems to global nonlinear controllability problems, Mathematical Control Theory, 14 (1985), 319-344.   Google Scholar

[26]

L. MalagutiS. Perrotta and V. Taddei, Exact controllability of infinite dimensional systems with controls of minimal norm, Topol. Methods Nonlinear Anal., 54 (2019), 1001-1021.  doi: 10.12775/tmna.2019.087.  Google Scholar

[27]

V. Obukhovskii and P. Zecca, Controllability for systems governed by semilinear differential inclusions in a Banach space with a noncompact semigroup, Nonlinear Anal., 70 (2009), 3424-3436.  doi: 10.1016/j.na.2008.05.009.  Google Scholar

[28]

A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Applied Mathematical Sciences, no. 44, Springer, New York, 1983. doi: 10.1007/978-1-4612-5561-1.  Google Scholar

[29]

M. F. Pinaud and H. R. Henríquez, Controllability of systems with a general nonlocal condition, J. Differential Equations, 269 (2020), 4609-4642.  doi: 10.1016/j.jde.2020.03.029.  Google Scholar

[30]

L. S. Pul'kina and A. E. Savenkova, A problem with a nonlocal, with respect to time, condition for multidimensional hyperbolic equations, Russian Math., 60 (2016), 33-43.   Google Scholar

[31]

A. E. Taylor and D. C. Lay, Introduction to Functional Analysis, John Wiley & Sons Inc, New York, 1980.  Google Scholar

[32]

R. Triggiani, A note on the lack of exact controllability for mild solutions in Banach spaces, SIAM J. Control Optim., 15 (1977), 407-411.  doi: 10.1137/0315028.  Google Scholar

[33]

V. Vijayakumar and R. Murugesu, Controllability for a class of second-order evolution differential inclusions without compactness, Appl. Anal., 98 (2019), 1367-1385.  doi: 10.1080/00036811.2017.1422727.  Google Scholar

[34]

I. I. Vrabie, $C_0$ Semigroups and Applications, North-Holland Mathematics Studies, no. 191, North-Holland Publishing Co., Amsterdam, 2003.  Google Scholar

[35]

J. Zabczyk, Mathematical Control Theory. An Introduction. Reprint of the 1995 Edition, Modern Birkhäuser Classics, Birkhäuser Boston Inc., Boston, MA, 2008. doi: 10.1007/978-0-8176-4733-9.  Google Scholar

[36]

E. Zuazua, An Introduction to the Exact Controllability for Distributed Systems, Textos e Notas, C.M.A.F., Universidades de Lisboa, 44 (1990). Google Scholar

[37]

E. Zuazua, Exact controllability for semilinear wave equations in one space dimension, Ann. Inst. H. Poincaré Anal. Non Linéaire, 10 (1993), 109–129. doi: 10.1016/S0294-1449(16)30221-9.  Google Scholar

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