Exact controllability and stability analysis for implicit fractional differential equations

Duman O.

Ukrainian Mathematical Journal, cilt.78, sa.9-10, ss.1-16, 2026 (Hakemli Dergi)

Özet

This paper establishes a comprehensive analysis for implicit fractional differential equations of the form

$$\mathcal{D}_{C}^\rho w(x)=f\left(x, w(x), \mathcal{D}_{C}^\rho w(x), u(x)\right)$$

with Caputo derivative $\mathcal{D}_{C}^\rho$ and control structure.

We address three fundamental problems under minimal and verifiable hypotheses: existence of solutions, exact controllability, and Ulam–Hyers stability. To handle the implicit structure, we establish an equivalence lemma that reformulates the system as a well-posed integral-functional equation, enabling the application of fixed-point theory. The main contributions are as follows: (i) proving existence and uniqueness of solutions via the Bielecki norm without imposing additional restrictions such as contraction constants; (ii) exact controllability is achieved through fixed-point methods, providing explicit constructions of control functions that drive the system to any desired terminal state; and (iii) Ulam–Hyers stability is derived with explicit error estimates, obtained directly using weighted norms without invoking additional assumptions. All results are unified under precise Lipschitz and growth conditions, ensuring a broad range of applicability.