Abstract
To characterize the ultra-slow diffusion processes with time-dependent logarithmical law attenuation, the distributed order fractional diffusion equation is needed. This paper discusses an analysis of approximate controllability from the exterior of distributed order fractional diffusion problem with the fractional Laplace operator subject to the non-zero exterior condition. We first establish some well-posedness results, such as the existence, uniqueness and regularity of the solutions allowing the weighted function
μ
that may be non-continuous. Especially, we show that the solutions can be represented by the series for the integral of a real-valued function. After giving the unique continuation property of the adjoint system, approximate controllability of the system is also included.