Optimal Fixed-Finite-Dimensional Compensator for Burgers’ Equation with Unbounded Input/Output Operators

Abstract

In this paper we consider the p)roblem of using reduced order dynamic compensators to control a class of nonlinear parabolic distributed parameter systems. We concentrate on a system wi)hl unbounded input and output operators governed by Burgers' equation. We use a linearized model to compute loworder-finite-dimensional control laws by minimizing certain energy functionals. We then apply these laws to the nonlinear model. Standard approaches to this p)roblem employ model/controller redurtion techniques in conjunction with LQG theory. The approach used here is based on the finite-dimensional Bernstein/Hyland optimal projection theory which yields a fixed-finite-order controller.

Numerical simulations for the stabilization and estimation problem of a semilinear partial differential equation

“…We can also consider the more realistic situation when we assume that u is not fully available to determine c. In this case we deal with the system (9) where the matrix C is as above and we now set y ob to represent a vector function in R m ob . The vector mappings ζ and η represent samples of white noise with symmetric covariance matrices Q d ∈ M n×n and R d ∈ M m ob ×m ob , respectively.…”

“…Up to the author's knowledge, a theoretical frame, general enough to cover the case that we are dealing with, was not established yet. Concerning the numerical analysis, up to now, only the one dimensional case has been treated [9].…”

“…The control and stabilization of solutions for models described by partial differential equations has been widely studied for both the linear and the nonlinear case (see [3,9,12,25]). In the particular field of fluid dynamics the stabilization by means of boundary control has been subject of research both from the theoretical [28][29][30] and numerical point of view [1,4,7] while having airflow applications in mind.…”

Necessary optimality conditions for optimal distributed and (Neumann) boundary control of Burgers equation

“…and y(0, ·) = y 0 (·) ∈ H, in which b(ϕ, ψ, ), the continuous trilinear form, is (1), we shall always mean the weak solution in the sense of (2).…”

“…An optimal control framework for the viscous Burgers equation is constructed in [5] and initial results for both distributed as well as boundary control (Dirichlet and Neumann) are presented using a continuous-adjoint formulation. Burns and Marrekchi in [2] investigate the optimal fixed-finite-dimensional compensator problem for the Burgers equation with unbounded input/output operators. Agarwal et al in [1] discuss the optimal and robust control of the Burgers equation with disturbance, in which the quadratic performance index is employed to find the optimal controller for the Burgers equation.…”