# American Institute of Mathematical Sciences

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  March 2021 Early access  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 and Related Fields, 2021, 11 (1) : 211-236. doi: 10.3934/mcrf.2020034
##### References:
 [1] V. R. Cabanillas, S. 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. [2] P. Cannarsa, P. 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. [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. [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. [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. [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). [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. [8] S. G. Gindikin and L. R. Volevich, Distributions and Convolution Equations, Gordon and Breach Sci. Publ., Philadelphia, 1992. [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. [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. [11] T. H. Koornwinder, R. Wong, R. 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. [12] J.-L. Lions, Optimal Control of Systems Governed by Partial Differential Equations, Springer, Berlin, 1971. [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. [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. [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. [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. [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. [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.

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##### References:
 [1] V. R. Cabanillas, S. 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. [2] P. Cannarsa, P. 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. [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. [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. [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. [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). [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. [8] S. G. Gindikin and L. R. Volevich, Distributions and Convolution Equations, Gordon and Breach Sci. Publ., Philadelphia, 1992. [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. [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. [11] T. H. Koornwinder, R. Wong, R. 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. [12] J.-L. Lions, Optimal Control of Systems Governed by Partial Differential Equations, Springer, Berlin, 1971. [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. [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. [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. [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. [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. [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.
The functions $u_l^n$
(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 $u = u_N$.
(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 $u = u_N$.
(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 $u = u_{N,l}$.
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
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