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We report on the dynamics of localized structures in an inhomogeneousSwift-Hohenberg model describing pattern formation in the transverse plane ofan optical cavity. This real order parameter equation is valid close to thesecond order critical point associated with bistability. The optical cavity isilluminated by an inhomogeneous spatial gaussian pumping beam, and subjected totime-delayed feedback. The gaussian injection beam breaks the translationalsymmetry of the system by exerting an attracting force on the localizedstructure. We show that the localized structure can be pinned to the center ofthe inhomogeneity, suppressing the delay-induced drift bifurcation that hasbeen reported in the particular case where the injection is homogeneous,assumming a continous wave operation. Under an inhomogeneous spatial pumpingbeam, we perform the stability analysis of localized solutions to identifydifferent instability regimes induced by time-delayed feedback. In particular,we predict the formation of two-arm spirals, as well as oscillating anddepinning dynamics caused by the interplay of an attracting inhomogeneity anddestabilizing time-delayed feedback. The transition from oscillating todepinning solutions is investigated by means of numerical continuationtechniques. Analytically, we use two approaches based on either an orderparameter equation, describing the dynamics of the localized structure in thevicinity of the Hopf bifurcation or an overdamped dynamics of a particle in apotential well generated by the inhomogeneity. In the later approach, thetime-delayed feedback acts as a driving force.

Delay-Induced Depinning of Localized Structures in a spatially inhomogeneous Swift-Hohenberg Model