Finite-Approximate Controllability of ν-Caputo Fractional Systems

This paper introduces a methodology for examining finite-approximate controllability in Hilbert spaces for linear/semilinear (Formula presented.) -Caputo fractional evolution equations. A novel criterion for achieving finite-approximate controllability in linear (Formula presented.) -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 (Formula presented.) -Caputo fractional evolution system, we establish sufficient conditions for achieving finite-approximative controllability in the semilinear (Formula presented.) -Caputo fractional evolution equation. These findings extend and build upon recent advancements in this field. The paper also explores applications to (Formula presented.) -Caputo fractional heat equations.