In this paper, we study the average controllability of a random heat equation, with the diffusivity serving as the random variable drawn from a general probability distribution. We show that the solutions of such random heat equations are both null and approximately controllable in average from an arbitrary open set of the domain and in an arbitrarily small time, recovering the known result when the random diffusivity is uniformly or exponentially distributed.

A parameter dependent control system is a system whose dynamics are governed by parameter dependent operators. Each unique parameter value corresponds to a specific realization of the system. The usefulness of such systems becomes clear when considering the problem of modelling physical processes, where due to the uncertainties and complexities involved, are difficult to perfectly model; thus it becomes natural to model them using parameter dependent coefficients. In particular, equations whose parameters are random can be used to model many uncertain physical processes (Soong, 1973).

An example of such a process is heat diffusion through an inhomogeneous material. Generally speaking, in order to control such systems one must use controls dependent on the parameter (see e.g. Johnson and Nerurkar, 1998, Masterkov and Rodina, 2007, Masterkov and Rodina, 2008 and references therein). Note that it is not always possible to control every realization of the system using a control independent of the parameter even in cases where the parameter is *known* (cf. Remark 1); one can instead make a robust compromise to controlling every realization of the system by controlling instead the average of the state with respect to the unknown parameter.

This problem was first introduced in Zuazua (2014). There, the problem was formulated and solved in the setting of finite-dimensional systems. In Lü and Zuazua (2016) the problem of averaged controllability was studied in the context of partial differential equations(PDEs). There, the authors focused on heat and Schrödinger equations with random parameters. Our focus will be on averaged controllability of the heat equation, where the diffusivity coefficient is unknown, i.e., the diffusivity is a random variable, with only its probability density function known. The treatment of this problem will follow the one presented in Lü and Zuazua (2016). The contribution of this paper is as follows: we extend the result of Lü and Zuazua (2016) to show both null and approximate controllability in average for a random heat equation when the diffusivity is a random variable with a *general probability distribution*. Additionally, we characterize the necessity of a non-zero diffusivity for achieving average controllability. (Read More)