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Proper orthogonal decomposition for reduced basis feedback controllers for parabolic equations. (English) Zbl 0964.93032

The authors consider the infinite horizon LQR (linear-quadratic regulator) problem and the LQG (linear-quadratic Gaussian) problem for the heat equation in space dimension 1 with 1-dimensional control,

$\frac{\partial y\left(t,x\right)}{\partial t}=\kappa \frac{{\partial }^{2}y\left(t,x\right)}{\partial {x}^{2}}+b\left(x\right)u\left(t\right),\phantom{\rule{1.em}{0ex}}u\left(t,0\right)=u\left(t,X\right)=0·$

In the LQR problem one minimizes a suitable quadratic functional of the state and the control over controls in ${L}^{2}\left(0,\infty \right);$ the optimal control $\overline{u}\left(t\right)$ is given by a feedback law involving the solution to an operator algebraic Riccati equation. The LQG problem only requires a state estimate instead of the full state, the feedback law similar to that for the LQR problem. The authors use finite element discretizations and apply the proper orthogonal decomposition method to two examples of input collections. In the first example, the proper orthogonal decomposition is directly applied to the finite element basis of linear $B$-splines; in the other, time snapshots are included. The objective is the design of low-order controllers.

##### MSC:
 93B40 Computational methods in systems theory 93C20 Control systems governed by PDE 93B52 Feedback control 65M50 Mesh generation and refinement (IVP of PDE)