Regional gradient controllability of sub-diffusion processes

By sub-diffusion systems we mean to replace the first order time derivative of normal diffusion system with a Riemann–Liouville time fractional derivative of order α∈(0,1)α∈(0,1). In this paper, we first introduce a new definition of regional gradient controllability for the sub-diffusion system with the control inputs appearing in the differential equations as distributed inputs, which recovers the usual definition of regional gradient controllability as α→1α→1. The sufficient and necessary conditions on actuators to achieve the gradient controllability in a given subregion of the whole domain are presented. An approach to guarantee the regional gradient controllability of the problems studied within the considered subregion using minimum energy control effort is presented. Several examples are presented in the end to illustrate the effectiveness of our results, where zone actuators, pointwise actuators or filament actuators are respectively considered.