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Control problems for semilinear neutral differential equations in Hilbert spaces.

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This study explores approximate controllability for neutral functional differential equations using fractional operators and Lipschitz continuity. It establishes existence and regularity of solutions under fewer restrictions, offering a new method for control system analysis.

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Area of Science:

  • Mathematics
  • Control Theory
  • Functional Analysis

Background:

  • Neutral functional differential equations (NFDEs) are crucial in modeling complex dynamical systems.
  • Investigating the controllability of NFDEs with unbounded operators is essential for practical applications.
  • Existing methods often impose stringent conditions on the operators and nonlinearities.

Purpose of the Study:

  • To establish approximate controllability for NFDEs with unbounded operators in Hilbert spaces.
  • To relax existing constraints on nonlinear terms and operator properties.
  • To demonstrate the existence and regularity of solutions for neutral control systems.

Main Methods:

  • Utilizing fractional powers of operators to analyze solution regularity.
  • Applying the concept of local Lipschitz continuity for nonlinear terms.
  • Developing a theoretical framework for approximate controllability in Hilbert spaces.

Main Results:

  • The study provides novel results on the regularity of solutions for NFDEs.
  • It establishes approximate controllability for neutral functional differential control systems.
  • The findings are achieved with significantly fewer restrictions compared to prior literature.

Conclusions:

  • The developed methods offer a more generalized approach to analyzing NFDEs.
  • The results contribute to a deeper understanding of control system behavior in infinite-dimensional spaces.
  • A practical example illustrates the applicability of the main theoretical findings.