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
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

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
[1]

Ovidiu Cârjă, Alina Lazu. On the minimal time null controllability of the heat equation. Conference Publications, 2009, 2009 (Special) : 143-150. doi: 10.3934/proc.2009.2009.143

[2]

Yalçin Sarol, Frederi Viens. Time regularity of the evolution solution to fractional stochastic heat equation. Discrete & Continuous Dynamical Systems - B, 2006, 6 (4) : 895-910. doi: 10.3934/dcdsb.2006.6.895

[3]

Antonio Greco, Antonio Iannizzotto. Existence and convexity of solutions of the fractional heat equation. Communications on Pure & Applied Analysis, 2017, 16 (6) : 2201-2226. doi: 10.3934/cpaa.2017109

[4]

Angkana Rüland, Mikko Salo. Quantitative approximation properties for the fractional heat equation. Mathematical Control & Related Fields, 2020, 10 (1) : 1-26. doi: 10.3934/mcrf.2019027

[5]

Víctor Hernández-Santamaría, Liliana Peralta. Some remarks on the Robust Stackelberg controllability for the heat equation with controls on the boundary. Discrete & Continuous Dynamical Systems - B, 2020, 25 (1) : 161-190. doi: 10.3934/dcdsb.2019177

[6]

Abdelaziz Khoutaibi, Lahcen Maniar. Null controllability for a heat equation with dynamic boundary conditions and drift terms. Evolution Equations & Control Theory, 2019, 0 (0) : 0-0. doi: 10.3934/eect.2020023

[7]

Jin Takahashi, Eiji Yanagida. Time-dependent singularities in the heat equation. Communications on Pure & Applied Analysis, 2015, 14 (3) : 969-979. doi: 10.3934/cpaa.2015.14.969

[8]

Umberto Biccari, Mahamadi Warma. Null-controllability properties of a fractional wave equation with a memory term. Evolution Equations & Control Theory, 2019, 0 (0) : 0-0. doi: 10.3934/eect.2020011

[9]

Larbi Berrahmoune. Constrained controllability for lumped linear systems. Evolution Equations & Control Theory, 2015, 4 (2) : 159-175. doi: 10.3934/eect.2015.4.159

[10]

Umberto Biccari. Boundary controllability for a one-dimensional heat equation with a singular inverse-square potential. Mathematical Control & Related Fields, 2019, 9 (1) : 191-219. doi: 10.3934/mcrf.2019011

[11]

Patrick Martinez, Judith Vancostenoble. Exact controllability in "arbitrarily short time" of the semilinear wave equation. Discrete & Continuous Dynamical Systems - A, 2003, 9 (4) : 901-924. doi: 10.3934/dcds.2003.9.901

[12]

Jesus Ildefonso Díaz, Jacqueline Fleckinger-Pellé. Positivity for large time of solutions of the heat equation: the parabolic antimaximum principle. Discrete & Continuous Dynamical Systems - A, 2004, 10 (1&2) : 193-200. doi: 10.3934/dcds.2004.10.193

[13]

Kazuhiro Ishige, Asato Mukai. Large time behavior of solutions of the heat equation with inverse square potential. Discrete & Continuous Dynamical Systems - A, 2018, 38 (8) : 4041-4069. doi: 10.3934/dcds.2018176

[14]

Dong-Ho Tsai, Chia-Hsing Nien. On space-time periodic solutions of the one-dimensional heat equation. Discrete & Continuous Dynamical Systems - A, 2019, 0 (0) : 0-0. doi: 10.3934/dcds.2020037

[15]

Nguyen Huy Tuan, Donal O'Regan, Tran Bao Ngoc. Continuity with respect to fractional order of the time fractional diffusion-wave equation. Evolution Equations & Control Theory, 2019, 0 (0) : 0-0. doi: 10.3934/eect.2020033

[16]

Ruiyang Cai, Fudong Ge, Yangquan Chen, Chunhai Kou. Regional gradient controllability of ultra-slow diffusions involving the Hadamard-Caputo time fractional derivative. Mathematical Control & Related Fields, 2020, 10 (1) : 141-156. doi: 10.3934/mcrf.2019033

[17]

Hideaki Takagi. Unified and refined analysis of the response time and waiting time in the M/M/m FCFS preemptive-resume priority queue. Journal of Industrial & Management Optimization, 2017, 13 (4) : 1945-1973. doi: 10.3934/jimo.2017026

[18]

Yutaka Sakuma, Atsushi Inoie, Ken’ichi Kawanishi, Masakiyo Miyazawa. Tail asymptotics for waiting time distribution of an M/M/s queue with general impatient time. Journal of Industrial & Management Optimization, 2011, 7 (3) : 593-606. doi: 10.3934/jimo.2011.7.593

[19]

Ping Lin. Feedback controllability for blowup points of semilinear heat equations. Discrete & Continuous Dynamical Systems - B, 2017, 22 (4) : 1425-1434. doi: 10.3934/dcdsb.2017068

[20]

Víctor Hernández-Santamaría, Luz de Teresa. Robust Stackelberg controllability for linear and semilinear heat equations. Evolution Equations & Control Theory, 2018, 7 (2) : 247-273. doi: 10.3934/eect.2018012

2018 Impact Factor: 0.925

Metrics

  • PDF downloads (73)
  • HTML views (31)
  • Cited by (0)

Other articles
by authors

[Back to Top]