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<p>In this work, we investigate the boundary controllability of time-discrete parabolic systems, uniformly with respect to the discretization parameter. To establish our main results, we adapt and extend the moment method first introduced by Fattorini and Russell, to the time-discrete setting. While this method has proven effective in the continuous framework and in the space-discrete case, its adaptation to time-discrete systems is challenging due to the fact that the more accurate results in the field are based on several complex analysis tools. To overcome this, we introduce a new alternative proof for constructing biorthogonals to (generalized) exponential functions in the continuous setting, based on the original proof by Fattorini and Russell which avoids the use of these tools, and taking ideas from the block moment method introduced by Benabdallah, Morancey and the first author of this work. We then manage to adapt this strategy to the discrete setting, enabling the construction and estimation of biorthogonal families for some time-discrete functions that play the same role as the exponentials at the discrete level. Our results show that these biorthogonals can be uniformly estimated for a finite portion of the spectrum, determined by the discretization parameter <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(\tau\)</EquationSource> </InlineEquation>, and that converges to the whole spectrum as <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(\tau\)</EquationSource> </InlineEquation> goes to zero. Using this tool, we prove a relaxed null-controllability result for discrete parabolic systems. It says that there exists a bounded sequence of time discrete controls that makes the solution reach a target at final time tending to zero exponentially fast as the discretization parameter <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(\tau\)</EquationSource> </InlineEquation> goes to zero. We will also study the effects of the existence of a minimal null-control time and how this phenomenon translates into the discrete world.</p>

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Boundary controllability of time-discrete parabolic systems: a moments method approach

  • 期刊论文

  • 出版模式：

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  • 发表日期：

    2025年11月18日

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  • Franck Boyer,
  • Víctor Hernández-Santamaría

摘要

In this work, we investigate the boundary controllability of time-discrete parabolic systems, uniformly with respect to the discretization parameter. To establish our main results, we adapt and extend the moment method first introduced by Fattorini and Russell, to the time-discrete setting. While this method has proven effective in the continuous framework and in the space-discrete case, its adaptation to time-discrete systems is challenging due to the fact that the more accurate results in the field are based on several complex analysis tools. To overcome this, we introduce a new alternative proof for constructing biorthogonals to (generalized) exponential functions in the continuous setting, based on the original proof by Fattorini and Russell which avoids the use of these tools, and taking ideas from the block moment method introduced by Benabdallah, Morancey and the first author of this work. We then manage to adapt this strategy to the discrete setting, enabling the construction and estimation of biorthogonal families for some time-discrete functions that play the same role as the exponentials at the discrete level. Our results show that these biorthogonals can be uniformly estimated for a finite portion of the spectrum, determined by the discretization parameter \(\tau\) , and that converges to the whole spectrum as \(\tau\) goes to zero. Using this tool, we prove a relaxed null-controllability result for discrete parabolic systems. It says that there exists a bounded sequence of time discrete controls that makes the solution reach a target at final time tending to zero exponentially fast as the discretization parameter \(\tau\) goes to zero. We will also study the effects of the existence of a minimal null-control time and how this phenomenon translates into the discrete world.