Abstract:
Optimal control problems are part of the area of PDE constrained
optimization, which has recently become a very active research area.
Problems with quadratic objective function and linear constraints play a
fundamental role in this field. In this case, it is possible to derive
optimal controls in feedback form which are extremely important, in
particular from an applications' point of view. In the stationary case,
an algebraic Riccati equation has to be solved.
We consider some recent research in the area of the numerical solution of algebraic Riccati equations (ARE) in the context with Newton's method. In the first part, we consider Newton's method in a Hilbert space setting and we formulate and prove mesh independence result of the Newton iterates for a family of discretizations. Applications are given in the field of optimal control of PDEs and of ODEs with delay terms. The second part deals with an inexact version of Newton's method which is particularly suitable for the numerical solution of large scale AREs. We are able to extend some of the monotone convergence results of Kleinman to the inexact version. These results are not covered by the classical inexact Newton convergence theorems. In the last part of the talk we address variants of Newton's method which are sometimes used for direct calculations of the feedback matrix. We show that some of these versions are inherently unstable and justify this claim with theoretical and numerical results.
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