Numerical approximations and regularizations of Riccati equations arising in hyperbolic dynamics with unbounded control operators

This paper provides an approximation theory for numerical computations of the solutions to algebraic Riccati equations arising in hyperbolic, boundary control problems. One of the difficulties in the approximation theory for Riccati equations is that many attractive numerical methods (such as standard finite elements) do not satisfy a “uniform stabilizability” condition, which is necessary for the stability of the approximate Riccati solutions. To deal with these problems, a regularizationapproximation technique, based on the introduction of special “artificial” terms to the dynamics of the original model, is proposed. The need for this regularization appears to be a distinct feature of hyperbolic (hyperbolic-like) equations, rather than parabolic (parabolic-like) problems where the smoothing effect of the dynamics is beneficial for the convergence and stability properties of approximate solutions to the associated Riccati equations (see [14]). The ultimate result demonstrates that the regularized, finite-dimensional feedback control yields near optimal performance and that the corresponding Riccati solution satisfies all the desired convergence properties. The general theory is illustrated by an example of a boundary control problem associated with the Kirchoff plate model. Some numerical results are provided for the given example.