Ulam-Hyers Stability with Explicit Bounds for Switched Integro-Differential Systems under Dwell-Time Switching (Global Journal of Sciences, 2025)
This paper establishes Ulam-Hyers stability for first-order switched Volterra integro-differential equations under average dwell time switching constraints. We analyze systems of the form:y′(t)=pσ(t)(t)y(t)+qσ(t)(t)+∫0tKσ(t)(t,s)y(s) ds,t∈[0,T]y′(t) = p_{\sigma(t)}(t)y(t) + q_{\sigma(t)}(t) + \int_0^t K_{\sigma(t)}(t, s)y(s) \, ds,\quad t \in [0, T]y′(t)=pσ(t)​(t)y(t)+qσ(t)​(t)+∫0t​Kσ(t)​(t,s)y(s)ds,t∈[0,T]where σ(t):[0,T]→{1,…,N}\sigma(t) : [0, T] → \{1, \dots, N\}σ(t):[0,T]→{1,…,N} is a switching signal with average dwell time τa>0\tau_a > 0τa​>0. Using successive approximation methods, we prove existence and uniqueness of solutions while deriving explicit error bounds for approximate solutions. The stability constantC=T(N0+T/τa)e(M1+M2T)TC = T(N_0 + T/\tau_a)e^{(M_1 + M_2T)T}C=T(N0​+T/τa​)e(M1​+M2​T)Tquantifies robustness by incorporating switching parameters, kernel bounds, and the time horizon. Our results extend classical integro-differential equation theory to switched systems with memory effects, providing a foundation for robust control design in applications such as power electronics, biological processes, and thermal management systems.
Keywords: Ulam-Hyers stability; Integro-Differential Equations; Switched Systems; Average Dwell Time; Robust Control