• algebraic Riccati equation
• optimal control
• residual

The thesis is about continuous-time algebraic Riccati equations A^T X + X A + Q - X G X = 0 that can be obtained from autonomous linear quadratic optimal control problems.
To quantify the accuracy of a computed solution of an algebraic Riccati equation, we introduce the optimality residual, that is a measure of residual linked to the original optimal control problem. We obtain a way to compute this residual in terms of the approximate solution and the problem data, up to first-order corrections.
Among the methods to solve algebraic Riccati equations we focus on Schur's method, which works on finding the stable invariant subspace of a suitable Hamiltonian matrix. Scaling this matrix before finding the stable invariant subspace, i.e. applying a change of basis, can result in an improved solution of the equation. We analyze how the optimality residual changes with scaling and we propose some scaling strategies that aim at reducing this residual. Numerical experiments are done in order to compare these strategies.