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<p>This work explores strong and exponential stabilization for semilinear systems using the decomposition method in a Banach state space, employing two distinct controls. By introducing an observability condition based on the semigroup solution and the parameter <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(\delta \ge 2\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>δ</mi> <mo>≥</mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation>, we demonstrate the strong stabilization of the considered system with the first control for <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(\delta \ge 2\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>δ</mi> <mo>≥</mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation>. Additionally, we provide generalized decay estimates that reinforce the theoretical results. A second control is applied for <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(\delta = 2\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>δ</mi> <mo>=</mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation> to achieve exponential stabilization using a modified observability condition, which simplifies the verification process without requiring explicit knowledge of the semigroup. This novel decomposition method is applied for the first time in this context and is extended to Banach spaces, broadening its applicability to a variety of systems.</p>

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A new decomposition method approach for strong and exponential stabilization of semilinear systems in banach spaces

  • 期刊论文

  • 发表日期：

    2025年12月15日

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  • Ayoub Cheddour,
  • Abdelhai Elazzouzi

摘要

This work explores strong and exponential stabilization for semilinear systems using the decomposition method in a Banach state space, employing two distinct controls. By introducing an observability condition based on the semigroup solution and the parameter \(\delta \ge 2\) δ ≥ 2 , we demonstrate the strong stabilization of the considered system with the first control for \(\delta \ge 2\) δ ≥ 2 . Additionally, we provide generalized decay estimates that reinforce the theoretical results. A second control is applied for \(\delta = 2\) δ = 2 to achieve exponential stabilization using a modified observability condition, which simplifies the verification process without requiring explicit knowledge of the semigroup. This novel decomposition method is applied for the first time in this context and is extended to Banach spaces, broadening its applicability to a variety of systems.