March  2021, 11(1): 211-236. doi: 10.3934/mcrf.2020034

Controllability problems for the heat equation on a half-axis with a bounded control in the Neumann boundary condition

1. 

B. Verkin Institute for Low Temperature Physics and Engineering, of the National Academy of Sciences of Ukraine, 47 Nauky Ave., Kharkiv, 61103, Ukraine, V.N. Karazin Kharkiv National University, 4 Svobody Sq., Kharkiv, 61077, Ukraine

2. 

B. Verkin Institute for Low Temperature Physics and Engineering, of the National Academy of Sciences of Ukraine, 47 Nauky Ave., Kharkiv, 61103, Ukraine

* Corresponding author: Larissa Fardigola

Received  September 2019 Revised  June 2020 Published  August 2020

In the paper, the problems of controllability and approximate controllability are studied for the control system $ w_t = w_{xx} $, $ w_x(0,\cdot) = u $, $ x>0 $, $ t\in(0,T) $, where $ u\in L^\infty(0,T) $ is a control. It is proved that each initial state of the system is approximately controllable to each target state in a given time $ T $. A necessary and sufficient condition for controllability in a given time $ T $ is obtained in terms of solvability of a Markov power moment problem. It is also shown that there is no initial state which is null-controllable in a given time $ T $. Orthogonal bases are constructed in $ H^1 $ and $ H_1 $. Using these bases, numerical solutions to the approximate controllability problem are obtained. The results are illustrated by examples.

Citation: Larissa Fardigola, Kateryna Khalina. Controllability problems for the heat equation on a half-axis with a bounded control in the Neumann boundary condition. Mathematical Control & Related Fields, 2021, 11 (1) : 211-236. doi: 10.3934/mcrf.2020034
References:
[1]

V. R. CabanillasS. B. De Menezes and E. Zuazua, Null controllability in unbounded domains for the semilinear heat equation with nonlinearities involving gradient terms, J. Optim. Theory Appl., 110 (2001), 245-264.  doi: 10.1023/A:1017515027783.  Google Scholar

[2]

P. CannarsaP. Martinez and J. Vancostenoble, Null controllability of the heat equation in unbounded domains by a finite measure control region, ESAIM Control Optim. Calc. Var., 10 (2004), 381-408.  doi: 10.1051/cocv:2004010.  Google Scholar

[3]

J. Darde and S. Ervedoza, On the reachable set for the one-dimensional heat equation, SIAM J. Control Optim., 56 (2018), 1692-1715.  doi: 10.1137/16M1093215.  Google Scholar

[4]

L. de Teresa and E. Zuazua, Approximate controllability of a semilinear heat equation in unbounded domains, Nonlinear Anal., 37 (1999), 1059-1090.  doi: 10.1016/S0362-546X(98)00085-6.  Google Scholar

[5]

V. Dhamo and F. Tröltzsch, Some aspects of reachability for parabolic boundary control problems with control constraints, Comput. Optim. Appl., 50 (2011), 75-110.  doi: 10.1007/s10589-009-9310-1.  Google Scholar

[6]

L. V. Fardigola, Transformation Operators and Influence Operators in Control Problems, Dr.Hab. Thesis, B. Verkin Institute for Low Temperature Physics and Engineering of the National Academy of Sciences of Ukraine, Kharkiv, 2016 (Ukrainian). Google Scholar

[7]

L. Fardigola and K. Khalina, Reachability and controllability problems for the heat equation on a half-axis, Zh. Mat. Fiz. Anal. Geom., 15 (2019), 57-78.  doi: 10.15407/mag15.01.057.  Google Scholar

[8]

S. G. Gindikin and L. R. Volevich, Distributions and Convolution Equations, Gordon and Breach Sci. Publ., Philadelphia, 1992.  Google Scholar

[9]

L. Gosse and O. Runberg, Resolution of the finite Markov moment problem, Comptes Rendus Mathematiques, 341 (2005), 775-789.  doi: 10.1016/j.crma.2005.10.009.  Google Scholar

[10]

U. W. Hochstrasser, Orthogonal palynomials, in Handbook of Mathematical Functions with Formulas Graphs and Mathematical Tables, (Eds. M. Abramowitz and I.A. Stegun), National Bureau of Standards, Applied Mathematics Series 55, Washington, DC, 1972, 771–802. Google Scholar

[11] T. H. KoornwinderR. WongR. Koekoek and R. F. Swarttouw, Orthogonal Polynomials, in NIST Handbook of Mathematical Functions, (eds. F.W.J. Olver, D.M. Lozier, F.F. Boisvert, and C.W. Clark) Cambridge University Press, 2010.   Google Scholar
[12]

J.-L. Lions, Optimal Control of Systems Governed by Partial Differential Equations, Springer, Berlin, 1971.  Google Scholar

[13]

S. Micu and E. Zuazua, On the lack of null controllability of the heat equation on the half-space, Port. Math. (N.S.), 58 (2001), 1-24.   Google Scholar

[14]

S. Micu and E. Zuazua, On the lack of null controllability of the heat equation on the half-line, Trans. Amer. Math. Soc., 353 (2001), 1635-1659.  doi: 10.1090/S0002-9947-00-02665-9.  Google Scholar

[15]

A. Munch and P. Pedregal, Numerical null controllability of the heat equation through a least squares and variational approach, European J. Appl. Math., 25 (2014), 277-306.  doi: 10.1017/S0956792514000023.  Google Scholar

[16]

S. S. Sener and M. Subasi, On a Neumann boundary control in a parabolic system, Bound. Value Probl., 2015 (2015), Article number: 166, Available from: https://doi.org/10.1186/s13661-015-0430-5. doi: 10.1186/s13661-015-0430-5.  Google Scholar

[17]

X. Zhang, A unified controllability/observability theory for some stochastic and deterministic partial differential equations, Proceedings of the International Congress of Mathematicians, Hyderabad, India, Ⅳ (2010), 3008–3034. doi: 10.1007/978-0-387-89488-1.  Google Scholar

[18]

E. Zuazua, Some problems and results on the controllability of partial differential equations, in Proceedings of the Second European Congress of Mathematics, Budapest, July 1996, Progress in Mathematics, 169, Birkhäuser Verlag, Basel, 276–311.  Google Scholar

show all references

References:
[1]

V. R. CabanillasS. B. De Menezes and E. Zuazua, Null controllability in unbounded domains for the semilinear heat equation with nonlinearities involving gradient terms, J. Optim. Theory Appl., 110 (2001), 245-264.  doi: 10.1023/A:1017515027783.  Google Scholar

[2]

P. CannarsaP. Martinez and J. Vancostenoble, Null controllability of the heat equation in unbounded domains by a finite measure control region, ESAIM Control Optim. Calc. Var., 10 (2004), 381-408.  doi: 10.1051/cocv:2004010.  Google Scholar

[3]

J. Darde and S. Ervedoza, On the reachable set for the one-dimensional heat equation, SIAM J. Control Optim., 56 (2018), 1692-1715.  doi: 10.1137/16M1093215.  Google Scholar

[4]

L. de Teresa and E. Zuazua, Approximate controllability of a semilinear heat equation in unbounded domains, Nonlinear Anal., 37 (1999), 1059-1090.  doi: 10.1016/S0362-546X(98)00085-6.  Google Scholar

[5]

V. Dhamo and F. Tröltzsch, Some aspects of reachability for parabolic boundary control problems with control constraints, Comput. Optim. Appl., 50 (2011), 75-110.  doi: 10.1007/s10589-009-9310-1.  Google Scholar

[6]

L. V. Fardigola, Transformation Operators and Influence Operators in Control Problems, Dr.Hab. Thesis, B. Verkin Institute for Low Temperature Physics and Engineering of the National Academy of Sciences of Ukraine, Kharkiv, 2016 (Ukrainian). Google Scholar

[7]

L. Fardigola and K. Khalina, Reachability and controllability problems for the heat equation on a half-axis, Zh. Mat. Fiz. Anal. Geom., 15 (2019), 57-78.  doi: 10.15407/mag15.01.057.  Google Scholar

[8]

S. G. Gindikin and L. R. Volevich, Distributions and Convolution Equations, Gordon and Breach Sci. Publ., Philadelphia, 1992.  Google Scholar

[9]

L. Gosse and O. Runberg, Resolution of the finite Markov moment problem, Comptes Rendus Mathematiques, 341 (2005), 775-789.  doi: 10.1016/j.crma.2005.10.009.  Google Scholar

[10]

U. W. Hochstrasser, Orthogonal palynomials, in Handbook of Mathematical Functions with Formulas Graphs and Mathematical Tables, (Eds. M. Abramowitz and I.A. Stegun), National Bureau of Standards, Applied Mathematics Series 55, Washington, DC, 1972, 771–802. Google Scholar

[11] T. H. KoornwinderR. WongR. Koekoek and R. F. Swarttouw, Orthogonal Polynomials, in NIST Handbook of Mathematical Functions, (eds. F.W.J. Olver, D.M. Lozier, F.F. Boisvert, and C.W. Clark) Cambridge University Press, 2010.   Google Scholar
[12]

J.-L. Lions, Optimal Control of Systems Governed by Partial Differential Equations, Springer, Berlin, 1971.  Google Scholar

[13]

S. Micu and E. Zuazua, On the lack of null controllability of the heat equation on the half-space, Port. Math. (N.S.), 58 (2001), 1-24.   Google Scholar

[14]

S. Micu and E. Zuazua, On the lack of null controllability of the heat equation on the half-line, Trans. Amer. Math. Soc., 353 (2001), 1635-1659.  doi: 10.1090/S0002-9947-00-02665-9.  Google Scholar

[15]

A. Munch and P. Pedregal, Numerical null controllability of the heat equation through a least squares and variational approach, European J. Appl. Math., 25 (2014), 277-306.  doi: 10.1017/S0956792514000023.  Google Scholar

[16]

S. S. Sener and M. Subasi, On a Neumann boundary control in a parabolic system, Bound. Value Probl., 2015 (2015), Article number: 166, Available from: https://doi.org/10.1186/s13661-015-0430-5. doi: 10.1186/s13661-015-0430-5.  Google Scholar

[17]

X. Zhang, A unified controllability/observability theory for some stochastic and deterministic partial differential equations, Proceedings of the International Congress of Mathematicians, Hyderabad, India, Ⅳ (2010), 3008–3034. doi: 10.1007/978-0-387-89488-1.  Google Scholar

[18]

E. Zuazua, Some problems and results on the controllability of partial differential equations, in Proceedings of the Second European Congress of Mathematics, Budapest, July 1996, Progress in Mathematics, 169, Birkhäuser Verlag, Basel, 276–311.  Google Scholar

Figure 1.  The functions $ u_l^n $
Figure 2.  (A)–(D): The controls $ u_N $ defined by (70). (E), (F): The influence of these controls on the end state of the solution to (6), (7) with $ W^T(x) = \frac{1}{\sqrt\pi} \int_0^T e^{-\frac{x^2}{4\xi}}\frac{2\xi-1}{\sqrt\xi}d\xi $ and $ u = u_N $.
Figure 3.  (A)–(D): The controls $ u_N $ defined by (70). (E), (F): The influence of these controls on the end state of the solution to (6), (7) with $ W^T(x) = \frac{1}{\sqrt\pi} \int_0^T e^{-\frac{x^2}{4\xi}}\frac{2(\xi-1)^2-1}{\sqrt\xi}d\xi $ and $ u = u_N $.
Figure 4.  (A), (B): The controls $ u_{N,l} $ defined by (70). (C), (D): The influence of these controls on the end state $ W_N^l $ of the solution to (6), (7) with $ W^T(x) = \cosh xe^{-\frac{x^2}{2}-\frac{1}{4}} $ and $ u = u_{N,l} $.
Table 1.  The estimates for $ \left\| W^T-W_N^l\right\|^1 $
$ \varepsilon_N^1 $ $ \varepsilon_{N,l}^2 $ $ \varepsilon_N^1+\varepsilon_{N,l}^2 $
$ N=3 $, $ l=100 $ 0.18666 0.12756 0.31422
$ N=3 $, $ l=200 $ 0.18666 0.05927 0.24593
$ N=4 $, $ l=150 $ 0.01535 0.08648 0.10183
$ N=4 $, $ l=400 $ 0.01535 0.03038 0.04573
$ \varepsilon_N^1 $ $ \varepsilon_{N,l}^2 $ $ \varepsilon_N^1+\varepsilon_{N,l}^2 $
$ N=3 $, $ l=100 $ 0.18666 0.12756 0.31422
$ N=3 $, $ l=200 $ 0.18666 0.05927 0.24593
$ N=4 $, $ l=150 $ 0.01535 0.08648 0.10183
$ N=4 $, $ l=400 $ 0.01535 0.03038 0.04573
[1]

V. Vijayakumar, R. Udhayakumar, K. Kavitha. On the approximate controllability of neutral integro-differential inclusions of Sobolev-type with infinite delay. Evolution Equations & Control Theory, 2021, 10 (2) : 271-296. doi: 10.3934/eect.2020066

[2]

Michael Schmidt, Emmanuel Trélat. Controllability of couette flows. Communications on Pure & Applied Analysis, 2006, 5 (1) : 201-211. doi: 10.3934/cpaa.2006.5.201

[3]

Dugan Nina, Ademir Fernando Pazoto, Lionel Rosier. Controllability of a 1-D tank containing a fluid modeled by a Boussinesq system. Evolution Equations & Control Theory, 2013, 2 (2) : 379-402. doi: 10.3934/eect.2013.2.379

[4]

Valery Y. Glizer. Novel Conditions of Euclidean space controllability for singularly perturbed systems with input delay. Numerical Algebra, Control & Optimization, 2021, 11 (2) : 307-320. doi: 10.3934/naco.2020027

[5]

Teddy Pichard. A moment closure based on a projection on the boundary of the realizability domain: 1D case. Kinetic & Related Models, 2020, 13 (6) : 1243-1280. doi: 10.3934/krm.2020045

[6]

Lars Grüne, Luca Mechelli, Simon Pirkelmann, Stefan Volkwein. Performance estimates for economic model predictive control and their application in proper orthogonal decomposition-based implementations. Mathematical Control & Related Fields, 2021  doi: 10.3934/mcrf.2021013

[7]

Yila Bai, Haiqing Zhao, Xu Zhang, Enmin Feng, Zhijun Li. The model of heat transfer of the arctic snow-ice layer in summer and numerical simulation. Journal of Industrial & Management Optimization, 2005, 1 (3) : 405-414. doi: 10.3934/jimo.2005.1.405

[8]

Tomáš Roubíček. An energy-conserving time-discretisation scheme for poroelastic media with phase-field fracture emitting waves and heat. Discrete & Continuous Dynamical Systems - S, 2017, 10 (4) : 867-893. doi: 10.3934/dcdss.2017044

[9]

Irena PawŃow, Wojciech M. Zajączkowski. Global regular solutions to three-dimensional thermo-visco-elasticity with nonlinear temperature-dependent specific heat. Communications on Pure & Applied Analysis, 2017, 16 (4) : 1331-1372. doi: 10.3934/cpaa.2017065

[10]

Enkhbat Rentsen, Battur Gompil. Generalized Nash equilibrium problem based on malfatti's problem. Numerical Algebra, Control & Optimization, 2021, 11 (2) : 209-220. doi: 10.3934/naco.2020022

[11]

Alexandr Mikhaylov, Victor Mikhaylov. Dynamic inverse problem for Jacobi matrices. Inverse Problems & Imaging, 2019, 13 (3) : 431-447. doi: 10.3934/ipi.2019021

[12]

Armin Lechleiter, Tobias Rienmüller. Factorization method for the inverse Stokes problem. Inverse Problems & Imaging, 2013, 7 (4) : 1271-1293. doi: 10.3934/ipi.2013.7.1271

[13]

Hildeberto E. Cabral, Zhihong Xia. Subharmonic solutions in the restricted three-body problem. Discrete & Continuous Dynamical Systems - A, 1995, 1 (4) : 463-474. doi: 10.3934/dcds.1995.1.463

[14]

Vladimir Georgiev, Sandra Lucente. Focusing nlkg equation with singular potential. Communications on Pure & Applied Analysis, 2018, 17 (4) : 1387-1406. doi: 10.3934/cpaa.2018068

[15]

Daoyin He, Ingo Witt, Huicheng Yin. On the strauss index of semilinear tricomi equation. Communications on Pure & Applied Analysis, 2020, 19 (10) : 4817-4838. doi: 10.3934/cpaa.2020213

[16]

Michel Chipot, Mingmin Zhang. On some model problem for the propagation of interacting species in a special environment. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020401

[17]

Fritz Gesztesy, Helge Holden, Johanna Michor, Gerald Teschl. The algebro-geometric initial value problem for the Ablowitz-Ladik hierarchy. Discrete & Continuous Dynamical Systems - A, 2010, 26 (1) : 151-196. doi: 10.3934/dcds.2010.26.151

[18]

Gloria Paoli, Gianpaolo Piscitelli, Rossanno Sannipoli. A stability result for the Steklov Laplacian Eigenvalue Problem with a spherical obstacle. Communications on Pure & Applied Analysis, 2021, 20 (1) : 145-158. doi: 10.3934/cpaa.2020261

[19]

Hailing Xuan, Xiaoliang Cheng. Numerical analysis and simulation of an adhesive contact problem with damage and long memory. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2781-2804. doi: 10.3934/dcdsb.2020205

[20]

Marco Ghimenti, Anna Maria Micheletti. Compactness results for linearly perturbed Yamabe problem on manifolds with boundary. Discrete & Continuous Dynamical Systems - S, 2021, 14 (5) : 1757-1778. doi: 10.3934/dcdss.2020453

2019 Impact Factor: 0.857

Metrics

  • PDF downloads (85)
  • HTML views (231)
  • Cited by (0)

Other articles
by authors

[Back to Top]