April  2020, 19(4): 1949-1978. doi: 10.3934/cpaa.2020086

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

Dedicated to professor Tomás Caraballo on the occasion of his 60th birthday

Received  May 2019 Revised  October 2019 Published  January 2020

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.

Citation: Umberto Biccari, Mahamadi Warma, Enrique Zuazua. Controllability of the one-dimensional fractional heat equation under positivity constraints. Communications on Pure & Applied Analysis, 2020, 19 (4) : 1949-1978. doi: 10.3934/cpaa.2020086
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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.  Google Scholar

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R. M. ColomboG. GuerraM. Herty and V. Schleper, Optimal control in networks of pipes and canals, SIAM J. Control Optim, 48 (2009), 2032-2050.  doi: 10.1137/080716372.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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R. FourerD. M. Gay and B. W. Kernighan, A modeling language for mathematical programming, Management Science, 36 (1990), 519-554.   Google Scholar

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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.  Google Scholar

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M. Kwaśnicki, Spectral analysis of subordinate Brownian motions in half-line, Studia Math., 206 (2011), 211-271.  doi: 10.4064/sm206-3-2.  Google Scholar

[26]

M. Kwaśnicki, Eigenvalues of the fractional Laplace operator in the interval, J. Funct. Anal., 262 (2012), 2379-2402.   Google Scholar

[27]

K. Le Balc'h, Global null-controllability and nonnegative-controllability of slightly superlinear heat equations, J. Math. Pures Appl., to appear Google Scholar

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T. LeonoriI. PeralA. 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.  Google Scholar

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[30]

J.-L. Lions and E. Magenes, Problèmes aux limites non homogènes et applications, Dunod, 1968.  Google Scholar

[31]

J. LohéacE. 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.  Google Scholar

[32]

D. Maity, M. Tucsnak and E. Zuazua, Controllability of a class of infinite dimensional systems with age structure, Submitted. Google Scholar

[33]

D. MaityM. 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.  Google Scholar

[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.  Google Scholar

[35]

A. MartinM. 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.  Google Scholar

[36]

S. MicuI. 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.  Google Scholar

[37]

D. Pighin and E. Zuazua, Controllability under positivity constraints of semilinear heat equations, Math. Control. Relat. Fields, 8 (2018), 935-964.   Google Scholar

[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.  Google Scholar

[39]

P. I. Richards, Shock waves on the highway, Operations Res., 4 (1956), 42-51.  doi: 10.1287/opre.4.1.42.  Google Scholar

[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.  Google Scholar

[41]

W. R. Schneider, Grey noise, in Stochastic Processes, Physics and Geometry (Ascona and Locarno, 1988), World Sci. Publ., Teaneck, NJ, 1990, 676-681.  Google Scholar

[42]

L. Schwartz, Étude des sommes d'exponentielles réelles, Hermann, Paris, 1943.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[48]

M. Warma and S. Zamorano, Null controllability from the exterior of a one-dimensional nonlocal heat equation, arXiv: 1811.10477. Google Scholar

[49]

E. Zuazua, Controllability of partial differential equations, 3ème cycle. Castro Urdiales, Espagne. Google Scholar

show all references

References:
[1]

DyCon Toolbox, https://deustotech.github.io/dycon-platform-documentation/, 2019., Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[4]

U. BiccariM. 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.  Google Scholar

[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.  Google Scholar

[6]

M. BonforteA. 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.  Google Scholar

[7]

P. Cannarsa and G. Floridia, Approximate multiplicative controllability for degenerate parabolic problems with Robin boundary conditions, Comm. Appl. Ind. Math., 2 (2011).  Google Scholar

[8]

P. CannarsaG. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[12]

R. M. ColomboG. GuerraM. Herty and V. Schleper, Optimal control in networks of pipes and canals, SIAM J. Control Optim, 48 (2009), 2032-2050.  doi: 10.1137/080716372.  Google Scholar

[13]

E. Di NezzaG. 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.  Google Scholar

[14]

A. A. DubkovB. 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.  Google Scholar

[15]

C. FabreJ.-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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[19]

R. FourerD. M. Gay and B. W. Kernighan, A modeling language for mathematical programming, Management Science, 36 (1990), 519-554.   Google Scholar

[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.  Google Scholar

[21] R. GlowinskiJ.-L. Lions and J. He, Exact and Approximate Controllability for Distributed Parameter Systems. A Numerical Approach, Cambridge University Press, 2008.  doi: 10.1017/S0962492900002452.  Google Scholar
[22]

R. GorenfloF. 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.  Google Scholar

[23]

N. HegoburuP. 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.  Google Scholar

[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.  Google Scholar

[25]

M. Kwaśnicki, Spectral analysis of subordinate Brownian motions in half-line, Studia Math., 206 (2011), 211-271.  doi: 10.4064/sm206-3-2.  Google Scholar

[26]

M. Kwaśnicki, Eigenvalues of the fractional Laplace operator in the interval, J. Funct. Anal., 262 (2012), 2379-2402.   Google Scholar

[27]

K. Le Balc'h, Global null-controllability and nonnegative-controllability of slightly superlinear heat equations, J. Math. Pures Appl., to appear Google Scholar

[28]

T. LeonoriI. PeralA. 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.  Google Scholar

[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.  Google Scholar

[30]

J.-L. Lions and E. Magenes, Problèmes aux limites non homogènes et applications, Dunod, 1968.  Google Scholar

[31]

J. LohéacE. 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.  Google Scholar

[32]

D. Maity, M. Tucsnak and E. Zuazua, Controllability of a class of infinite dimensional systems with age structure, Submitted. Google Scholar

[33]

D. MaityM. 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.  Google Scholar

[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.  Google Scholar

[35]

A. MartinM. 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.  Google Scholar

[36]

S. MicuI. 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.  Google Scholar

[37]

D. Pighin and E. Zuazua, Controllability under positivity constraints of semilinear heat equations, Math. Control. Relat. Fields, 8 (2018), 935-964.   Google Scholar

[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.  Google Scholar

[39]

P. I. Richards, Shock waves on the highway, Operations Res., 4 (1956), 42-51.  doi: 10.1287/opre.4.1.42.  Google Scholar

[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.  Google Scholar

[41]

W. R. Schneider, Grey noise, in Stochastic Processes, Physics and Geometry (Ascona and Locarno, 1988), World Sci. Publ., Teaneck, NJ, 1990, 676-681.  Google Scholar

[42]

L. Schwartz, Étude des sommes d'exponentielles réelles, Hermann, Paris, 1943.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[48]

M. Warma and S. Zamorano, Null controllability from the exterior of a one-dimensional nonlocal heat equation, arXiv: 1811.10477. Google Scholar

[49]

E. Zuazua, Controllability of partial differential equations, 3ème cycle. Castro Urdiales, Espagne. Google Scholar

Figure 1.  Graphic of the function $ q(x) $
Figure 2.  Evolution in the time interval $ (0,T_{\rm min}) $ of the solution of (2.1) with $ s = 0.8 $. The blue curve is the target we want to reach while the green bullets indicate the target we computed numerically
Figure 8.  Evolution in the time interval $ (0,T_{\rm min}) $ of the solution of (2.1) with $ s = 0.8 $. The blue curve is the target we want to reach while the green bullets indicate the target we computed numerically
Figure 3.  Minimal-time control: space-time distribution of the impulses. The white lines delimit the control region $ \omega = (-0.3,0.8) $. The regions in which the control is active are marked in yellow
Figure 4.  Minimal-time control: intensity of the impulses in logarithmic scale. In the $ (t,x) $ plane in blue the time $ t $ varies from $ t = 0 $ (left) to $ t = T_{\rm min} $ (right)
Figure 10.  Minimal-time control: intensity of the impulses in logarithmic scale. In the $ (t,x) $ plane in blue the time $ t $ varies from $ t = 0 $ (left) to $ t = T_{\rm min} $ (right)
Figure 5.  Evolution in the time interval $ (0,0.9) $ of the solution of (2.1) with $ s = 0.8 $. The blue curve is the target we want to reach while the green bullets indicate the target we computed numerically
Figure 6.  Behavior of the control in time $ T = 0.9 $. The white lines delimit the control region $ \omega = (-0.3,0.8) $. The regions in which the control is active are marked in yellow. The atomic nature is lost
Figure 7.  Evolution in the time interval $ (0,0.7) $ of the solution of (2.1) with $ s = 0.8 $ (left) and of the control $ u $ (right), under the constraint $ u\geq 0 $. The bold characters highlight the control region $ \omega = (-0.3,0.8) $. The control remains inactive during the entire time interval, and the equation is not controllable
Figure 9.  Minimal-time control: space-time distribution of the impulses. The white lines delimit the control region $ \omega = (-0.3,0.8) $. The regions in which the control is active are marked in yellow
Figure 11.  Evolution in the time interval $ (0,0.4) $ of the solution of (2.1) with $ s = 0.8 $. The blue curve is the target we want to reach while the green bullets indicate the target we computed numerically
Figure 12.  Behavior of the control in time $ T = 0.4 $. The white lines delimit the control region $ \omega = (-0.3,0.8) $. The regions in which the control is active are marked in yellow. The atomic nature is lost
Figure 13.  Evolution in the time interval $ (0,0.15) $ of the solution of (2.1) with $ s = 0.8 $ (left) and of the control $ u $ (right), under the constraint $ u\geq 0 $. The bold characters highlight the control region $ \omega = (-0.3,0.8) $. The equation is not controllable
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