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

  • * Corresponding author: Larissa Fardigola

    * Corresponding author: Larissa Fardigola 
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  • 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.

    Mathematics Subject Classification: Primary:93B05, 35K05;Secondary:35B30.


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  • 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
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