Performance Degradation and Stabilization of Semilinear Stochastic Systems Driven by α-Stable Lévy Noise

Classical stochastic control theory predominantly relies on Gaussian noise models driven by continuous Wiener processes. However, these models fail to capture the heavy-tailed, discontinuous jump phenomena frequently observed in modern complex systems, such as power grids under fault conditions or networks experiencing sudden cyber-anomalies. This paper investigates the finite-time stabilization and approximate controllability of a class of semilinear impulsive stochastic evolution equations defined on a separable Hilbert space and subjected to non-Gaussian α-stable Lévy noise.

When the perturbation paradigm shifts from continuous diffusion to discontinuous jump processes, standard feedback control strategies designed under strict second-moment assumptions often experience severe performance degradation or total loss of stability. Because the variance of the Lévy process diverges for α∈(1,2), classical mean-square L² stability analysis entirely collapses. To address this fundamental limitation, we shift our analytical framework to the Banach space L^p(Ω, H) of fractional-moment integrable processes, where the moments remain finite for 1<p<α.

Within this space, we establish finite-approximate controllability by formulating a parameterized feedback control law and constructing a nonlinear operator mapping based on the mild solution of the impulsive system. By employing Picard iterations and the Banach fixed-point principle under the p-th moment norm, we theoretically guarantee the existence of a unique stabilizing state trajectory that compensates for both continuous nonlinear drift and discrete impulsive shocks.

To complement the theoretical framework, we present a preliminary numerical investigation to assess controller degradation explicitly. Through extensive simulations utilizing the Chambers–Mallows–Stuck method to generate symmetric α-stable increments, we demonstrate the failure modes of classical controllers under heavy-tailed perturbations. Ultimately, this work illustrates how control adjustments based on fractional moments facilitate finite-time stabilization, establishing a mathematically rigorous foundation for the synthesis of resilient controllers in non-Gaussian, impulsive stochastic environments.