  Non-Hermitian models describe the physics of ubiquitous open systems with
gain and loss. One intriguing aspect of non-Hermitian models is their inherent
topology that can produce intriguing boundary phenomena like resilient
higher-order topological insulators (HOTIs) and non-Hermitian skin effects
(NHSE). Recently, time-multiplexed lattices in synthetic dimensions have
emerged as a versatile platform for the investigation of these effects free of
geometric restrictions. Despite holding broad applications, studies of these
effects have been limited to static cases so far, and full dynamical control
over the non-Hermitian effects has remained elusive. Here, we demonstrate the
emergence of topological non-Hermitian corner states with remarkable temporal
controllability and robustness in a two-dimensional photonic synthetic time
lattice. Specifically, we showcase various dynamic control mechanisms for light
confinement and flow, including spatial mode tapering, sequential
non-Hermiticity on-off switching, dynamical corner state relocation, and light
steering. Moreover, we establish the corner state's robustness in the presence
of intensity modulation randomness and quantitatively determine its breakdown
regime. Our findings extend non-Hermitian and topological photonic effects into
higher synthetic dimensions, offering remarkable flexibility and real-time
control possibilities. This opens avenues for topological classification,
quantum walk simulations of many-body dynamics, and robust Floquet engineering,
free from the limitations of physical geometries.
