On average controllability of random heat equations with arbitrarily distributed diffusivity
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)