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Controllability of the one-dimensional fractional heat equation under positivity constraints
1. | Chair of Computational Mathematics, Fundación Deusto, Av. de las Universidades 24, 48007 Bilbao, Basque Country, Spain |
2. | Facultad de Ingeniería, Universidad de Deusto, Av. de las Universidades 24, 48007 Bilbao, Basque Country, Spain |
3. | George Mason University, Department of Mathematical Sciences, Fairfax, VA 22030, USA |
4. | Chair in Applied Analysis, Alexander von Humboldt-Professorship, Department of Mathematics, Friedrich-Alexander-Universität, Erlangen-Nürnberg, 91058 Erlangen, Germany |
5. | Departamento de Matemáticas, Universidad Autonóma de Madrid, 24049, Madrid, Spain |
In this paper, we analyze the controllability properties under positivity constraints on the control or the state of a one-dimensional heat equation involving the fractional Laplacian $ (-d_x^{\,2})^{s}{} $ ($ 0<s<1 $) on the interval $ (-1,1) $. We prove the existence of a minimal (strictly positive) time $ T_{\rm min} $ such that the fractional heat dynamics can be controlled from any initial datum in $ L^2(-1,1) $ to a positive trajectory through the action of a positive control, when $ s>1/2 $. Moreover, we show that in this minimal time constrained controllability is achieved by means of a control that belongs to a certain space of Radon measures. We also give some numerical simulations that confirm our theoretical results.
References:
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DyCon Toolbox, https://deustotech.github.io/dycon-platform-documentation/, 2019., |
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W. Arendt, C. J. K. Batty, M. Hieber and F. Neubrander, Vector-valued Laplace transforms and Cauchy problems, vol. 96 of Monographs in Mathematics, 2nd edition
doi: 10.1007/978-3-0348-0087-7. |
[3] |
U. Biccari and V. Hernández-Santamaría,
Controllability of a one-dimensional fractional heat equation: Theoretical and numerical aspects, IMA J. Math. Control. Inf., 36.4 (2019), 1199-1235.
doi: 10.1109/TAC.1985.1103850. |
[4] |
U. Biccari, M. Warma and E. Zuazua,
Local elliptic regularity for the Dirichlet fractional Laplacian, Adv. Nonlinear Stud., 17 (2017), 387-409.
doi: 10.1515/ans-2017-0014. |
[5] |
U. Biccari, M. Warma and E. Zuazua, Local regularity for fractional heat equations, in Recent Advances in PDEs: Analysis, Numerics and Control, Springer, 2018, 233-249. |
[6] |
M. Bonforte, A. Figalli and X. Ros-Oton,
Infinite speed of propagation and regularity of solutions to the fractional porous medium equation in general domains, Comm. Pure Appl. Math., 70 (2017), 1472-1508.
doi: 10.1002/cpa.21673. |
[7] |
P. Cannarsa and G. Floridia, Approximate multiplicative controllability for degenerate parabolic problems with Robin boundary conditions, Comm. Appl. Ind. Math., 2 (2011). |
[8] |
P. Cannarsa, G. Floridia and A. Y. Khapalov,
Multiplicative controllability for semilinear reaction-diffusion equations with finitely many changes of sign, J. Math. Pures Appl., 108 (2017), 425-458.
doi: 10.1016/j.matpur.2017.07.002. |
[9] |
P. Cannarsa and A. Y. Khapalov,
Multiplicative controllability for reaction-diffusion equations with target states admitting finitely many changes of sign, Discrete Contin. Dyn. Syst. Ser. B, 14 (2010), 1293-1311.
doi: 10.3934/dcdsb.2010.14.1293. |
[10] |
W. L. Chan and B. Z. Guo,
Optimal birth control of population dynamics. Ⅱ. Problems with free final time, phase constraints, and mini-max costs, J. Math. Anal. Appl., 146 (1990), 523-539.
doi: 10.1016/0022-247X(90)90322-7. |
[11] |
R. M. Colombo and A. Groli,
Minimising stop and go waves to optimise traffic flow, Appl. Math. Letters, 17 (2004), 697-701.
doi: 10.1016/S0893-9659(04)90107-3. |
[12] |
R. M. Colombo, G. Guerra, M. Herty and V. Schleper,
Optimal control in networks of pipes and canals, SIAM J. Control Optim, 48 (2009), 2032-2050.
doi: 10.1137/080716372. |
[13] |
E. Di Nezza, G. Palatucci and E. Valdinoci,
Hitchhiker's guide to the fractional Sobolev spaces, Bull. Sci. Math., 136 (2012), 521-573.
doi: 10.1016/j.bulsci.2011.12.004. |
[14] |
A. A. Dubkov, B. Spagnolo and V. V. Uchaikin,
Lévy flight superdiffusion: An introduction, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 18 (2008), 2649-2672.
doi: 10.1142/S0218127408021877. |
[15] |
C. Fabre, J.-P. Puel and E. Zuazua,
Approximate controllability of the semilinear heat equation, Proc. Roy. Soc. Edinburgh Sec. A Math., 125 (1995), 31-61.
doi: 10.1017/S0308210500030742. |
[16] |
E. Fernandez-Cara and A. M{ü}nch,
Numerical exact controllability of the 1d heat equation: duality and Carleman weights, J. Optim. Theor. Appl., 163 (2014), 253-285.
doi: 10.1007/s10957-013-0517-z. |
[17] |
E. Fernández-Cara and E. Zuazua,
Null and approximate controllability for weakly blowing up semilinear heat equations, Ann. I. H. Poincare Nonlin. Anal., 17 (2000), 583-616.
doi: 10.1016/S0294-1449(00)00117-7. |
[18] |
G. Floridia,
Approximate controllability for nonlinear degenerate parabolic problems with bilinear control, J. Differential Equations, 257 (2014), 3382-3422.
doi: 10.1016/j.jde.2014.06.016. |
[19] |
R. Fourer, D. M. Gay and B. W. Kernighan,
A modeling language for mathematical programming, Management Science, 36 (1990), 519-554.
|
[20] |
O. Glass,
On the controllability of the 1-d isentropic euler equation, J. Eur. Math. Soc., 9 (2007), 427-486.
doi: 10.4171/JEMS/85. |
[21] |
R. Glowinski, J.-L. Lions and J. He, Exact and Approximate Controllability for Distributed Parameter Systems. A Numerical Approach, Cambridge University Press, 2008.
doi: 10.1017/S0962492900002452.![]() ![]() ![]() |
[22] |
R. Gorenflo, F. Mainardi and A. Vivoli,
Continuous-time random walk and parametric subordination in fractional diffusion, Chaos Solitons Fractals, 34 (2007), 87-103.
doi: 10.1016/j.chaos.2007.01.052. |
[23] |
N. Hegoburu, P. Magal and M. Tucsnak,
Controllability with positivity constraints of the Lotka-McKendrick system, SIAM J. Control Optim., 56 (2018), 723-750.
doi: 10.1137/16M1103087. |
[24] |
V. Keyantuo and M. Warma,
On the interior approximate controllability for fractional wave equations, Discrete Contin. Dyn. Syst., 36 (2016), 3719-3739.
doi: 10.3934/dcds.2016.36.3719. |
[25] |
M. Kwaśnicki,
Spectral analysis of subordinate Brownian motions in half-line, Studia Math., 206 (2011), 211-271.
doi: 10.4064/sm206-3-2. |
[26] |
M. Kwaśnicki,
Eigenvalues of the fractional Laplace operator in the interval, J. Funct. Anal., 262 (2012), 2379-2402.
|
[27] |
K. Le Balc'h, Global null-controllability and nonnegative-controllability of slightly superlinear heat equations, J. Math. Pures Appl., to appear |
[28] |
T. Leonori, I. Peral, A. Primo and F. Soria,
Basic estimates for solutions of a class of nonlocal elliptic and parabolic equations, Discrete Contin. Dyn. Syst., 35 (2015), 6031-6068.
doi: 10.3934/dcds.2015.35.6031. |
[29] |
M. J. Lighthill and G. B. Whitham,
On kinematic waves Ⅱ. a theory of traffic flow on long crowded roads, Proc. Roy. Soc. London. Series A Math. Phys. Sci., 229 (1955), 317-345.
doi: 10.1098/rspa.1955.0089. |
[30] |
J.-L. Lions and E. Magenes, Problèmes aux limites non homogènes et applications, Dunod, 1968. |
[31] |
J. Lohéac, E. Trélat and E. Zuazua,
Minimal controllability time for the heat equation under unilateral state or control constraints, Math. Models Methods Appl. Sci., 27 (2017), 1587-1644.
doi: 10.1142/S0218202517500270. |
[32] |
D. Maity, M. Tucsnak and E. Zuazua, Controllability of a class of infinite dimensional systems with age structure, Submitted. |
[33] |
D. Maity, M. Tucsnak and E. Zuazua,
Controllability and positivity constraints in population dynamics with age structuring and diffusion, J. Math. Pures Appl., 129 (2019), 153-179.
doi: 10.1016/j.matpur.2018.12.006. |
[34] |
B. B. Mandelbrot and J. W. Van Ness,
Fractional Brownian motions, fractional noises and applications, SIAM Rev., 10 (1968), 422-437.
doi: 10.1137/1010093. |
[35] |
A. Martin, M. Möller and S. Moritz,
Mixed integer models for the stationary case of gas network optimization, Math. Prog., 105 (2006), 563-582.
doi: 10.1007/s10107-005-0665-5. |
[36] |
S. Micu, I. Roventa and M. Tucsnak,
Time optimal boundary controls for the heat equation, J. Funct. Anal., 263 (2012), 25-49.
doi: 10.1016/j.jfa.2012.04.009. |
[37] |
D. Pighin and E. Zuazua,
Controllability under positivity constraints of semilinear heat equations, Math. Control. Relat. Fields, 8 (2018), 935-964.
|
[38] |
D. Pighin and E. Zuazua, Controllability under positivity constraints of multi-d wave equations, in Trends in Control Theory and Partial Differential Equations, Springer, 2019, 195-232. |
[39] |
P. I. Richards,
Shock waves on the highway, Operations Res., 4 (1956), 42-51.
doi: 10.1287/opre.4.1.42. |
[40] |
D. A. Rüland,
Unique continuation for fractional schrödinger equations with rough potentials, Comm. Partial Differential Equations, 40 (2015), 77-114.
doi: 10.1080/03605302.2014.905594. |
[41] |
W. R. Schneider, Grey noise, in Stochastic Processes, Physics and Geometry (Ascona and Locarno, 1988), World Sci. Publ., Teaneck, NJ, 1990, 676-681. |
[42] |
L. Schwartz, Étude des sommes d'exponentielles réelles, Hermann, Paris, 1943. |
[43] |
R. Servadei and E. Valdinoci,
On the spectrum of two different fractional operators, Proc. Roy. Soc. Edinburgh Sect. A, 144 (2014), 831-855.
doi: 10.1017/S0308210512001783. |
[44] |
M. C. Steinbach,
On pde solution in transient optimization of gas networks, J. Comput. Appl. Math., 203 (2007), 345-361.
doi: 10.1016/j.cam.2006.04.018. |
[45] |
A. Wächter and L. T. Biegler,
On the implementation of an interior-point filter line-search algorithm for large-scale nonlinear programming, Math. Program., 106 (2006), 25-57.
doi: 10.1007/s10107-004-0559-y. |
[46] |
M. Warma,
The fractional relative capacity and the fractional Laplacian with Neumann and Robin boundary conditions on open sets, Potential Anal., 42 (2015), 499-547.
doi: 10.1007/s11118-014-9443-4. |
[47] |
M. Warma,
Approximate controllabilty from the exterior of space-time fractional diffusive equations, SIAM J. Control Optim., 57 (2019), 2037-2063.
doi: 10.1137/18M117145X. |
[48] |
M. Warma and S. Zamorano, Null controllability from the exterior of a one-dimensional nonlocal heat equation, arXiv: 1811.10477. |
[49] |
E. Zuazua, Controllability of partial differential equations, 3ème cycle. Castro Urdiales, Espagne. |
show all references
References:
[1] |
DyCon Toolbox, https://deustotech.github.io/dycon-platform-documentation/, 2019., |
[2] |
W. Arendt, C. J. K. Batty, M. Hieber and F. Neubrander, Vector-valued Laplace transforms and Cauchy problems, vol. 96 of Monographs in Mathematics, 2nd edition
doi: 10.1007/978-3-0348-0087-7. |
[3] |
U. Biccari and V. Hernández-Santamaría,
Controllability of a one-dimensional fractional heat equation: Theoretical and numerical aspects, IMA J. Math. Control. Inf., 36.4 (2019), 1199-1235.
doi: 10.1109/TAC.1985.1103850. |
[4] |
U. Biccari, M. Warma and E. Zuazua,
Local elliptic regularity for the Dirichlet fractional Laplacian, Adv. Nonlinear Stud., 17 (2017), 387-409.
doi: 10.1515/ans-2017-0014. |
[5] |
U. Biccari, M. Warma and E. Zuazua, Local regularity for fractional heat equations, in Recent Advances in PDEs: Analysis, Numerics and Control, Springer, 2018, 233-249. |
[6] |
M. Bonforte, A. Figalli and X. Ros-Oton,
Infinite speed of propagation and regularity of solutions to the fractional porous medium equation in general domains, Comm. Pure Appl. Math., 70 (2017), 1472-1508.
doi: 10.1002/cpa.21673. |
[7] |
P. Cannarsa and G. Floridia, Approximate multiplicative controllability for degenerate parabolic problems with Robin boundary conditions, Comm. Appl. Ind. Math., 2 (2011). |
[8] |
P. Cannarsa, G. Floridia and A. Y. Khapalov,
Multiplicative controllability for semilinear reaction-diffusion equations with finitely many changes of sign, J. Math. Pures Appl., 108 (2017), 425-458.
doi: 10.1016/j.matpur.2017.07.002. |
[9] |
P. Cannarsa and A. Y. Khapalov,
Multiplicative controllability for reaction-diffusion equations with target states admitting finitely many changes of sign, Discrete Contin. Dyn. Syst. Ser. B, 14 (2010), 1293-1311.
doi: 10.3934/dcdsb.2010.14.1293. |
[10] |
W. L. Chan and B. Z. Guo,
Optimal birth control of population dynamics. Ⅱ. Problems with free final time, phase constraints, and mini-max costs, J. Math. Anal. Appl., 146 (1990), 523-539.
doi: 10.1016/0022-247X(90)90322-7. |
[11] |
R. M. Colombo and A. Groli,
Minimising stop and go waves to optimise traffic flow, Appl. Math. Letters, 17 (2004), 697-701.
doi: 10.1016/S0893-9659(04)90107-3. |
[12] |
R. M. Colombo, G. Guerra, M. Herty and V. Schleper,
Optimal control in networks of pipes and canals, SIAM J. Control Optim, 48 (2009), 2032-2050.
doi: 10.1137/080716372. |
[13] |
E. Di Nezza, G. Palatucci and E. Valdinoci,
Hitchhiker's guide to the fractional Sobolev spaces, Bull. Sci. Math., 136 (2012), 521-573.
doi: 10.1016/j.bulsci.2011.12.004. |
[14] |
A. A. Dubkov, B. Spagnolo and V. V. Uchaikin,
Lévy flight superdiffusion: An introduction, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 18 (2008), 2649-2672.
doi: 10.1142/S0218127408021877. |
[15] |
C. Fabre, J.-P. Puel and E. Zuazua,
Approximate controllability of the semilinear heat equation, Proc. Roy. Soc. Edinburgh Sec. A Math., 125 (1995), 31-61.
doi: 10.1017/S0308210500030742. |
[16] |
E. Fernandez-Cara and A. M{ü}nch,
Numerical exact controllability of the 1d heat equation: duality and Carleman weights, J. Optim. Theor. Appl., 163 (2014), 253-285.
doi: 10.1007/s10957-013-0517-z. |
[17] |
E. Fernández-Cara and E. Zuazua,
Null and approximate controllability for weakly blowing up semilinear heat equations, Ann. I. H. Poincare Nonlin. Anal., 17 (2000), 583-616.
doi: 10.1016/S0294-1449(00)00117-7. |
[18] |
G. Floridia,
Approximate controllability for nonlinear degenerate parabolic problems with bilinear control, J. Differential Equations, 257 (2014), 3382-3422.
doi: 10.1016/j.jde.2014.06.016. |
[19] |
R. Fourer, D. M. Gay and B. W. Kernighan,
A modeling language for mathematical programming, Management Science, 36 (1990), 519-554.
|
[20] |
O. Glass,
On the controllability of the 1-d isentropic euler equation, J. Eur. Math. Soc., 9 (2007), 427-486.
doi: 10.4171/JEMS/85. |
[21] |
R. Glowinski, J.-L. Lions and J. He, Exact and Approximate Controllability for Distributed Parameter Systems. A Numerical Approach, Cambridge University Press, 2008.
doi: 10.1017/S0962492900002452.![]() ![]() ![]() |
[22] |
R. Gorenflo, F. Mainardi and A. Vivoli,
Continuous-time random walk and parametric subordination in fractional diffusion, Chaos Solitons Fractals, 34 (2007), 87-103.
doi: 10.1016/j.chaos.2007.01.052. |
[23] |
N. Hegoburu, P. Magal and M. Tucsnak,
Controllability with positivity constraints of the Lotka-McKendrick system, SIAM J. Control Optim., 56 (2018), 723-750.
doi: 10.1137/16M1103087. |
[24] |
V. Keyantuo and M. Warma,
On the interior approximate controllability for fractional wave equations, Discrete Contin. Dyn. Syst., 36 (2016), 3719-3739.
doi: 10.3934/dcds.2016.36.3719. |
[25] |
M. Kwaśnicki,
Spectral analysis of subordinate Brownian motions in half-line, Studia Math., 206 (2011), 211-271.
doi: 10.4064/sm206-3-2. |
[26] |
M. Kwaśnicki,
Eigenvalues of the fractional Laplace operator in the interval, J. Funct. Anal., 262 (2012), 2379-2402.
|
[27] |
K. Le Balc'h, Global null-controllability and nonnegative-controllability of slightly superlinear heat equations, J. Math. Pures Appl., to appear |
[28] |
T. Leonori, I. Peral, A. Primo and F. Soria,
Basic estimates for solutions of a class of nonlocal elliptic and parabolic equations, Discrete Contin. Dyn. Syst., 35 (2015), 6031-6068.
doi: 10.3934/dcds.2015.35.6031. |
[29] |
M. J. Lighthill and G. B. Whitham,
On kinematic waves Ⅱ. a theory of traffic flow on long crowded roads, Proc. Roy. Soc. London. Series A Math. Phys. Sci., 229 (1955), 317-345.
doi: 10.1098/rspa.1955.0089. |
[30] |
J.-L. Lions and E. Magenes, Problèmes aux limites non homogènes et applications, Dunod, 1968. |
[31] |
J. Lohéac, E. Trélat and E. Zuazua,
Minimal controllability time for the heat equation under unilateral state or control constraints, Math. Models Methods Appl. Sci., 27 (2017), 1587-1644.
doi: 10.1142/S0218202517500270. |
[32] |
D. Maity, M. Tucsnak and E. Zuazua, Controllability of a class of infinite dimensional systems with age structure, Submitted. |
[33] |
D. Maity, M. Tucsnak and E. Zuazua,
Controllability and positivity constraints in population dynamics with age structuring and diffusion, J. Math. Pures Appl., 129 (2019), 153-179.
doi: 10.1016/j.matpur.2018.12.006. |
[34] |
B. B. Mandelbrot and J. W. Van Ness,
Fractional Brownian motions, fractional noises and applications, SIAM Rev., 10 (1968), 422-437.
doi: 10.1137/1010093. |
[35] |
A. Martin, M. Möller and S. Moritz,
Mixed integer models for the stationary case of gas network optimization, Math. Prog., 105 (2006), 563-582.
doi: 10.1007/s10107-005-0665-5. |
[36] |
S. Micu, I. Roventa and M. Tucsnak,
Time optimal boundary controls for the heat equation, J. Funct. Anal., 263 (2012), 25-49.
doi: 10.1016/j.jfa.2012.04.009. |
[37] |
D. Pighin and E. Zuazua,
Controllability under positivity constraints of semilinear heat equations, Math. Control. Relat. Fields, 8 (2018), 935-964.
|
[38] |
D. Pighin and E. Zuazua, Controllability under positivity constraints of multi-d wave equations, in Trends in Control Theory and Partial Differential Equations, Springer, 2019, 195-232. |
[39] |
P. I. Richards,
Shock waves on the highway, Operations Res., 4 (1956), 42-51.
doi: 10.1287/opre.4.1.42. |
[40] |
D. A. Rüland,
Unique continuation for fractional schrödinger equations with rough potentials, Comm. Partial Differential Equations, 40 (2015), 77-114.
doi: 10.1080/03605302.2014.905594. |
[41] |
W. R. Schneider, Grey noise, in Stochastic Processes, Physics and Geometry (Ascona and Locarno, 1988), World Sci. Publ., Teaneck, NJ, 1990, 676-681. |
[42] |
L. Schwartz, Étude des sommes d'exponentielles réelles, Hermann, Paris, 1943. |
[43] |
R. Servadei and E. Valdinoci,
On the spectrum of two different fractional operators, Proc. Roy. Soc. Edinburgh Sect. A, 144 (2014), 831-855.
doi: 10.1017/S0308210512001783. |
[44] |
M. C. Steinbach,
On pde solution in transient optimization of gas networks, J. Comput. Appl. Math., 203 (2007), 345-361.
doi: 10.1016/j.cam.2006.04.018. |
[45] |
A. Wächter and L. T. Biegler,
On the implementation of an interior-point filter line-search algorithm for large-scale nonlinear programming, Math. Program., 106 (2006), 25-57.
doi: 10.1007/s10107-004-0559-y. |
[46] |
M. Warma,
The fractional relative capacity and the fractional Laplacian with Neumann and Robin boundary conditions on open sets, Potential Anal., 42 (2015), 499-547.
doi: 10.1007/s11118-014-9443-4. |
[47] |
M. Warma,
Approximate controllabilty from the exterior of space-time fractional diffusive equations, SIAM J. Control Optim., 57 (2019), 2037-2063.
doi: 10.1137/18M117145X. |
[48] |
M. Warma and S. Zamorano, Null controllability from the exterior of a one-dimensional nonlocal heat equation, arXiv: 1811.10477. |
[49] |
E. Zuazua, Controllability of partial differential equations, 3ème cycle. Castro Urdiales, Espagne. |













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