Boundary stabilization in finite time of one-dimensional linear hyperbolic balance laws with coefficients depending on time and space

In this article we are interested in the boundary stabilization finite time of one-dimensional linear hyperbolic balance laws with coefficients depending on and space. We extend so called "backstepping method" by introducing appropriate time-dependent integral transformations order to map our initial system a new one which has desired stability properties. The kernels involved solutions non standard multi-dimensional PDEs, where dependence introduces several difficulties treatment their well-posedness. This work generalizes previous results literature, only time-independent systems were considered.