Finite‐time boundary stabilization of reaction‐diffusion systems

Abstract

In this paper, the problem of boundary finite-time stabilization is considered for reaction-diffusion systems (RDSs). First, a full-domain controller is designed, and sufficient conditions are given to ensure finite-time stability of RDSs under the designed controller. Then, for practical applications, a boundary controller is designed to obtain finite-time stability. By virtue of the finite-time stability lemma, criteria are presented to guarantee the finite-time stability of RDSs for the Neumann boundary conditions and the mixed boundary conditions. As an extension to uncertain RDSs, robust finite-time stabilization is studied, and criterion is obtained under the boundary control. Numerical simulations verify the effectiveness of the proposed design techniques. KEYWORDS boundary control, finite-time stability, full-domain control, reaction-diffusion systems, uncertain reaction-diffusion systems Recently, the research of finite-time stability for partial differential systems has also attracted some interest. 19,20 The reaction-diffusion system (RDS) has been widely used to characterize the system dynamics found in applications in the chemical processes, fluid flows, and biological pattern formation (see previous works 21-28 for a survey). The stabilization of RDSs has attracted significant interest. 21,[24][25][26] For RDSs, there exists a specific control strategy and a boundary control. [29][30][31][32][33] While the boundary stabilization for RDSs has been well investigated, 34,35 only a few reported the finite-time boundary stabilization for RDSs. This paper focuses on the finite-time boundary stabilization for RDSs. How the boundary controller can be designed remains a main challenge since there exist a few results available for use. To provide the necessary theoretical framework in the study, we first derive a full-domain controller for reaching finite-time stability of RDSs. Making use of the finite-time stability lemma and Wirtinger's inequality, we present sufficient conditions to guarantee the finite-time stability under Int J Robust Nonlinear Control. 2018;28:1641-1652.wileyonlinelibrary.com/journal/rnc

Boundary control of stochastic reaction‐diffusion systems with Markovian switching

“…Lemma 1 (Wirtinger's inequality 32 ). Let z be a vector function with z(0) = 0 or z(1) = 0, and z ∈ W 1,2 ([0, 1]; R n ).…”

“…It is observed that system (31) is not asymptotically stable with u(t) = 0. Taking account into the consideration of boundary controller (32), the operation mode and initial value are taken the same as before. In Figure 2A, the state norm E||y(⋅, t)|| 2 reaches to zero.…”

Finite-Time Speed Control of Marine Diesel Engine Based on ADRC

“…is new type of ESO, which is established by inverse hyperbolic sine function, has fewer adjustment parameters and simple theoretical analysis. On the basis of feedback linearization of the ESO, the sliding mode error feedback control law is designed based on the theory of finite-time convergence [24]. According to Lyapunov stability theory, the control law can converge in finite time.…”

Boundary control of linear stochastic reaction‐diffusion systems

“…In this study, we investigate the boundary control problem of linear SRDSs with Neumann boundary conditions. Boundary control, which requires the actuator to be placed at the boundary of the system domain, is an appropriate approach to enable an SRDS to achieve certain specified performance requirements . In the work of Pan et al, a boundary control was used to achieve the mean‐square asymptotical stability for SRDSs with Neumann boundary conditions, but the result in the aforementioned work is conservative because only one single point's state was used to design the boundary controller.…”

“…Boundary control, which requires the actuator to be placed at the boundary of the system domain, is an appropriate approach to enable an SRDS to achieve certain specified performance requirements. [13][14][15][16] In the work of Pan et al, 13 a boundary control was used to achieve the mean-square asymptotical stability for SRDSs with Neumann boundary conditions, but the result in the aforementioned work 13 is conservative because only one single point's state was used to design the boundary controller. As a classical method for studying boundary control of deterministic systems, the backstepping method, until recently, can only be used to study deterministic linear systems.…”