Closed-Form Boundary State Feedbacks for a Class of 1-D Partial Integro-Differential Equations

Abstract

Abstract-In this paper, a problem of boundary stabilization of a class of linear parabolic partial integro-differential equations (P(I)DEs) in one dimension is considered using the method of backstepping, avoiding spatial discretization required in previous efforts. The problem is formulated as a design of an integral operator whose kernel is required to satisfy a hyperbolic P(I)DE. The kernel P(I)DE is then converted into an equivalent integral equation and by applying the method of successive approximations, the equation's well posedness and the kernel's smoothness are established. It is shown how to extend this approach to design optimally stabilizing controllers. An adaptation mechanism is developed to reduce the conservativeness of the inverse optimal controller, and the performance bounds are derived. For a broad range of physically motivated special cases feedback laws are constructed explicitly and the closed-loop solutions are found in closed form. A numerical scheme for the kernel P(I)DE is proposed; its numerical effort compares favorably with that associated with operator Riccati equations.

Control for fast and stable Laminar-to-High-Reynolds-Numbers transfer in a 2D Navier-Stokes channel flow

“…One has to solve a finite set of linear PIDE equations (19)- (21) for computing the K n 's; we provide an symbolically computable solution via a convergent infinite series, whose partial sums provides an approximation to the controller. The kernel equations can be solved numerically as well, which can be done fast and efficiently compared, for example, with LQR-where nonlinear time dependent Riccati equations appear (see [37] for a numerical comparison between LQR and backstepping).…”

“…The following definitions establish facts and notations useful for designing our control laws, based on the backstepping method (see [37]). This method consists in finding an invertible transformation of the original variables into others whose stability properties are easy to establish.…”

“…Consider the pressure terms like those in the second line of (37). Using the divergencefree condition iγ n u n + v ny = 0, and integrating by parts,…”

“…This structural property is a sort of "spatial causality", requiring that in the expression for, say, f (t, y), no value of f (t, s) for s > y appears. It is a technical requirement in the backstepping method for parabolic equations (see [37,39]) used next. Seeking the strict-feedback structure in (42), we choose V n such thaṫ…”

Backstepping observer design for parabolic PDEs with measurement of weighted spatial averages

“…The proposed framework is based on the infinite dimensional backstepping approach (Balogh and Krstic, 2002;Liu, 2003;Smyshlyaev and Krstic, 2004), which is a systematic design tool for state feedback gains. The observer gain is determined so that the error system is converted into an exponentially stable target system by a state transformation called the backstepping transformation.…”