Finite-Approximate Controllability of <mml:semantics>ν</mml:semantics>-Caputo Fractional Systems

This paper introduces a methodology for examining finite-approximate controllability in Hilbert spaces for linear/semilinear nu-Caputo fractional evolution equations. A novel criterion for achieving finite-approximate controllability in linear nu-Caputo fractional evolution equations is established, utilizing resolvent-like operators. Additionally, we identify a control strategy that not only satisfies the approximative controllability property but also ensures exact finite-dimensional controllability. Leveraging the approximative controllability of the corresponding linear nu-Caputo fractional evolution system, we establish sufficient conditions for achieving finite-approximative controllability in the semilinear nu-Caputo fractional evolution equation. These findings extend and build upon recent advancements in this field. The paper also explores applications to nu-Caputo fractional heat equations.