$ L^p $- null controllability of an abstract differential inclusion with a nonlocal condition

Bholanath Kumbhakar; Dwijendra Narain Pandey

doi:10.3934/mcrf.2025001

Article Contents

2025, Volume 15, Issue 3: 1121-1149. Doi: 10.3934/mcrf.2025001

This issue Previous Article Convergence of minimal norm control problems of linear heat equations with time delay Next Article On a new concept of controllability of second-order semilinear differential equations in Banach spaces

$ L^p $- null controllability of an abstract differential inclusion with a nonlocal condition

Bholanath Kumbhakar and
Dwijendra Narain Pandey^,

Department of Mathematics, Indian Institute of Technology Roorkee, Roorkee, India

^*Corresponding author: Dwijendra Narain Pandey
^*Corresponding author: Dwijendra Narain Pandey

Received: April 2024

Revised: October 2024

Early access: January 10, 2025

Published online: January 10, 2025

The first author is supported by Council of Scientific and Industrial Research, New Delhi, Government of India (File No. 09/143(0954)/2019-EMR-I).

Abstract / Introduction Full Text(HTML) Related Papers Cited by

Abstract

In this paper, we discuss the $ L^p $- null controllability of the abstract semilinear differential inclusion with nonlocal conditions. The control function $ u $ takes value in $ L^p(I, U) $, $ 1<p<\infty $, $ I = [0, \nu] $ and the admissible controls set $ U $ is a uniformly convex Banach space. By presuming null controllability for the linear system with source term, we apply an approximate solvability technique to reduce the problem to finite-dimensional subspaces. Consequently, the solutions for the primary problem are the limiting functions within these finite dimensional subspaces. The paper offers a unique solution to a challenge introduced by assuming $ U $ as a uniformly convex Banach space, which presents issues of convexity during the construction of the necessary control. Such issues are not present when $ U $ is a separable Hilbert space. Therefore, the paper's novelty is its successful resolution of the convexity problem, paving the way for $ L^p(I, U) $ null controllability of the semilinear differential control system in which $ U $ is a uniformly convex Banach space.

Keywords:

Mathematics Subject Classification: Primary: 34A60, 93B05; Secondary: 34H05.

Citation:

Full Text(HTML)

Related Papers

Cited by

References

[1]	B. Ainseba, Exact and approximate controllability of the age and space population dynamics structured model, J. Math. Anal. Appl., 275 (2002), 562-574. doi: 10.1016/S0022-247X(02)00238-X.
[2]	S. Aniţa, Analysis and Control of Age-Dependent Population Dynamics, Math. Model. Theory Appl., 11. Kluwer Academic Publishers, Dordrecht, 2000.
[3]	J. P. Aubin and A. Cellina, Differential Inclusions: Set-Valued Maps and Viability Theory, Grundlehren Math. Wiss., 264. Springer-Verlag, Berlin, 1984.
[4]	I. Benedetti, V. Obukhovskii and P. Zecca, Controllability for impulsive semilinear functional differential inclusions with a non-compact evolution operator, Discuss. Math. Differ. Incl. Control Optim., 31 (2011), 39-69. doi: 10.7151/dmdico.1127.
[5]	I. Benedetti, L. Malaguti and V. Taddei, Semilinear evolution equations in abstract spaces and applications, Rend. Istit. Mat. Univ. Trieste, 44 (2012), 371-388.
[6]	I. Benedetti, V. Taddei and M. Väth, Evolution problems with nonlinear nonlocal boundary conditions, J. Dynam. Differential Equations, 25 (2013), 477-503. doi: 10.1007/s10884-013-9303-8.
[7]	I. Benedetti, V. 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.
[8]	I. Cioranescu, Geometry of Banach Spaces, Duality Mappings and Nonlinear Problems, Mathematics and its Applications, Vol. 62, Kluwer Academic Publishers Group, Dordrecht, 1990.
[9]	R. F. Curtain and A. J. Pritchard, Infinite Dimensional Linear Systems Theory, Lecture Notes in Control and Information Sciences, 8. Springer-Verlag, Berlin-New York, 1978.
[10]	J. P. Dauer and N. I. Mahmudov, Exact null controllability of semilinear integrodifferential systems in Hilbert spaces, J. Math. Anal. Appl., 299 (2004), 322-332. doi: 10.1016/j.jmaa.2004.01.050.
[11]	K. Deimling, Multivalued Differential Equations, De Gruyter Ser. Nonlinear Anal. Appl., 1. Walter de Gruyter & Co., Berlin, 1992.
[12]	K. Deng, Exponential decay of solutions of semilinear parabolic equations with nonlocal initial conditions, J. Math. Anal. Appl., 179 (1993), 630-637. doi: 10.1006/jmaa.1993.1373.
[13]	I. Ekeland and R. Temam, Convex Analysis and Variational Problems, Studies in Mathematics and its Applications, Vol. 1, North-Holland Publishing Co., Amsterdam-Oxford; American Elsevier Publishing Co., Inc., New York, 1976.
[14]	D. Gallaun, C. Seifert and M. Tautenhahn, Sufficient criteria and sharp geometric conditions for observability in Banach spaces, SIAM J. Control Optim., 58 (2020), 2639-2657. doi: 10.1137/19M1266769.
[15]	S. Hu and N. S. Papageorgiou, Handbook of Multivalued Analysis. Vol. I, Mathematics and its Applications, 419. Kluwer Academic Publishers, Dordrecht, 1997.
[16]	N. Hegoburu and M. Tucsnak, Null controllability of the Lotka-McKendrick system with spatial diffusion, Math. Control Relat. Fields, 8 (2018), 707-720. doi: 10.3934/mcrf.2018030.
[17]	M. Kamenskii, V. Obukhovskii and P. Zecca, Condensing Multivalued Maps and Semilinear Differential Inclusions in Banach Spaces, De Gruyter Series in Nonlinear Analysis and Applications, 7. Walter de Gruyter & Co., Berlin, 2001.
[18]	O. Kovrijkine, Some results related to the Logvinenko-Sereda theorem, Proc. Amer. Math. Soc., 129 (2001), 3037-3047. doi: 10.1090/S0002-9939-01-05926-3.
[19]	J.-C. Louis and D. Wexler, On exact controllability in Hilbert spaces, J. Differential Equations, 49 (1983), 258-269. doi: 10.1016/0022-0396(83)90014-1.
[20]	N. I. Mahmudov, Exact null controllability of semilinear evolution systems, J. Global Optim., 56 (2013), 317-326. doi: 10.1007/s10898-011-9823-x.
[21]	L. Malaguti and P. Rubbioni, Nonsmooth feedback controls of nonlocal dispersal models conditions, Nonlinearity, 29 (2016), 823-850. doi: 10.1088/0951-7715/29/3/823.
[22]	L. Malaguti, S. 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.
[23]	L. Malaguti, S. Perrotta and V. Taddei, $L^p$-exact controllability of partial differential equations with nonlocal terms, Evol. Equ. Control Theory, 11 (2022), 1533-1564. doi: 10.3934/eect.2021053.
[24]	S. Migórski and A. Ochal, Quasi-static hemivariational inequality via vanishing acceleration approach, SIAM J. Math. Anal., 41 (2009), 1415-1435. doi: 10.1137/080733231.
[25]	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.
[26]	M. F. Pinaud and H. R. Henriquez, Controllability of systems with a general nonlocal condition, J. Differential Equations, 269 (2020), 4609-4642. doi: 10.1016/j.jde.2020.03.029.
[27]	Y. Simporé, Null controllability of a nonlinear population dynamics with age structuring and spatial diffusion, Nonlinear Analysis, Geometry and Applications, Birkhäuser/Springer, Cham, (2020), 1-33. doi: 10.1007/978-3-030-57336-2_1.
[28]	G. Tenenbaum and M. Tucsnak, On the null-controllability of diffusion equations, ESAIM Control Optim. Calc. Var., 17 (2011), 1088-1100. doi: 10.1051/cocv/2010035.
[29]	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.
[30]	A. Vieru, On null controllability of linear systems in Banach spaces, Systems Control Lett., 54 (2005), 331-337. doi: 10.1016/j.sysconle.2004.09.004.
[31]	Y.-Y. Yu and F.-Z. Wang, Solvability for a nonlocal dispersal model governed by time and space integrals, Open Math., 20 (2022), 1785-1799. doi: 10.1515/math-2022-0552.