Exact controllability and stability analysis for implicit fractional differential equations

Abstract

UDC 517.9; 517.97

We present a comprehensive analysis of implicit fractional-differential equations of the form $$\mathcal{D}_{C}^\rho w(x)=f\big(x, w(x), \mathcal{D}_{C}^\rho w(x), u(x)\big)$$ 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 the Ulam–Hyers stability. To handle the implicit structure, we establish an equivalence lemma that reformulates the system as a well-posed integral-functional equation, which enables the application of fixed-point theory. The main contributions are as follows:  (i) proving the 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) the Ulam–Hyers stability is derived with explicit error estimates obtained directly by using weighted norms without invoking additional assumptions. All results are unified under precise Lipschitz and growth conditions, ensuring a broad range of applicability.