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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 and Control Theory, doi: 10.3934/eect.2021053
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.

[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.

[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.

[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.

[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.

[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.

[7]

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

[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.

[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.

[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.

[11]

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

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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.

[13]

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

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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.

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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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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. 

[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.

[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.

[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.

[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.

[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. 

[31]

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

[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.

[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.

[34]

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

[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.

[36]

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

[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.

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.

[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.

[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.

[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.

[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.

[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.

[7]

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

[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.

[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.

[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.

[11]

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

[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.

[13]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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. 

[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.

[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.

[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.

[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.

[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. 

[31]

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

[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.

[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.

[34]

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

[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.

[36]

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

[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.

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