STABILIZATION OF A PARABOLIC PDE SANDWICHED BY TWO NONLINEAR ODES

This paper is devoted to the stabilization of a class of nonlinear parabolic ODE-PDE-ODE systems constituting by a parabolic equation sandwiched by two nonlinear ODEs. Different from the related literature where both the two ODE subsystems are linear time invariant (LTI) or only the ODE subsystem proximal to the control input is nonlinear but that distal to the input is LTI, serious nonlinearities are contained in the system under investigation since both the two ODE subsystems (no matter proximal or distal to the input) are all nonlinear which lead to the incapability of the control schemes on this topic. To solve the control problem, a novel control framework is established by smartly combining infinite-dimensional backstepping method with the the finite-dimensional one. Specifically, three steps of backstepping transformations are subsequently introduced for the system, which include two finite-dimensional ones respectively for the distal and proximal ODE subsystems and an infinite-dimensional one for the PDE subsystem. Then, a new target system is obtained under the backstepping transformations while a state-feedback controller is explicitly designed. Finally, by recursive analysis from the target system, desirable stability of the resulting closed-loop system is obtained, i.e., all the states of the resulting closed-loop system are bounded and converge to zero ultimately. A simulation example is provided to validate the effectiveness of the proposed theoretical results.