Radially varying delay‐compensated distributed control of reaction‐diffusion PDEs on n‐ball under revolution symmetry conditions

Abstract

In this article, a distributed control design is developed via partial differential equation (PDE) backstepping for a class of reaction-diffusion PDEs on n-ball under revolution symmetry conditions in the presence of radial spatially varying actuator delay. The actuator delay is modeled by a two-dimensional (2D) first-order hyperbolic PDE with spatially varying boundary, resulting in a cascaded transport-diffusion PDE system. Two-step backstepping transformation is introduced for the control design. First, a new Volterra transformation for the n-ball under revolution symmetry is proposed that allows the kernel function to be solvable. Different from the kernel in the 1D domain, which is expressed by the classical Fourier series, the kernel of the n-ball under revolution symmetry conditions is based on the Fourier-Bessel series. The resulting Volterra transformation is an implicit one that contains the state of the target system on both sides of the definition. Based on the implicit transformation, we employ the successive approximations approach to derive the second-step transformation, namely, an explicit transformation, and arrive at a stable target system. The inverse transformation is also derived, to prove L 2 exponential stability of the closed-loop system.The stability analysis requires more technical skills that are not needed in solving the spatially varying delay compensation problem when the domain is 1D.Numerical simulation examples of 3D and 4D revolution-symmetrical systems illustrate the effectiveness of the controller.

Intermittent boundary control for robust fixed‐time stabilization of uncertain reaction‐diffusion systems

“…. That is, lim t→ T ||𝜓(t, ⋅)|| = 0 holds and ||𝜓(t, ⋅)|| = 0 is true for any t ≥ T. Thence, system (6) achieves robust FxTS via the aperiodically intermittent boundary controller (4). ▪…”

“…In addition, we give Corollary 4 to ensure that system (33) can achieve FxTS when 𝜉 = 0. (33) achieves FxTS via the boundary controller (4)…”

“…[1][2][3] Contrary to systems characterized by ordinary differential equations, the dynamic evolution of RDSs depends on both time and space variables and is able to describe the main characteristics and development patterns of objects. [4][5][6] At present, RDSs have been widely employed in many scientific fields including chemical engineering, fluid mechanics, and population ecology. 7 Stability is the essential prerequisite for the normal operation of RDSs.…”

“…Reaction‐diffusion systems (RDSs), an important class of parabolic partial differential systems, originate from reaction‐diffusion phenomena that are widespread in the objective world, such as the spread of infectious diseases in biological populations, fluid motion in porous media, and heat conduction in physics 1‐3 . Contrary to systems characterized by ordinary differential equations, the dynamic evolution of RDSs depends on both time and space variables and is able to describe the main characteristics and development patterns of objects 4‐6 . At present, RDSs have been widely employed in many scientific fields including chemical engineering, fluid mechanics, and population ecology 7 …”