### Conveners

#### Day II: Session I

- Valeria Simoncini (Universita di Bologna)

#### Day II: Session II

- Beatrice Meini ()

#### Day II: Session III

- Jens Saak (Max Planck Institute for Dynamics of Complex Technical Systems)

We consider the nonsymmetric T-Riccati equation

$$ 0 = \mathcal{R}_T(X):=DX+X^TA-X^TBX+C,\qquad (1) $$ where $A,B,C,D\in\mathbb{R}^{n\times n}$ and sufficient conditions for the existence and uniqueness of a minimal (w.r.t. entry-wise comparison) solution $X_{\min}\in\mathbb{R}^{n\times n}$ are provided. To date, the nonlinear matrix equation (1) is still an unexplored problem in...

The numerical solution of the algebraic Riccati matrix equation

$$(1)\quad A^T X + XA − XSX + Q = 0,$$
where $A$, $S$, $Q\in\mathbb{R}^{n\times n}$ , is an interesting and still challenging task especially when the problem dimension is very large, say $n > 10^4$ , as the dense solution $X$ cannot be store and a memory-saving approximation has to be sought.

A vast portion of the...

Our goal is the feedback stabilization of a two-dimensional two-phase Stefan problem coupled with (Navier-)Stokes equations. The Stefan problem can model solidification and melting of pure materials and gets its name from the purely algebraic Stefan condition which describes the coupling between the temperature of the material and its melting process.

After linearization and discretization,...

We consider the numerical solution of large-scale, differential matrix Riccati equations (DRE). Projection methods have recently arisen as a promising class of solution strategies. Existing approaches in this class focus on polynomial or extended Krylov subspaces as approximation space. We show that great computational and memory advantages are obtained with fully rational Krylov subspaces and...

We consider the differential Riccati equation,

$$\dot{X} = A^T X + X A - X BB^T X + C^T C.$$

The differential Riccati equation as well as the algebraic Riccati equation play important roles in applied mathematics like control theory and system theory. In our talk, we focus on the large-scale case. The numerical solution of these equation is challenging, in particular, because of the...

Matrix equations have arisen as the natural setting for various PDE discretization methods such as finite differences, isogeometric analysis, spectral and finite elements.

Thanks to major recent computational advances, solving certain classes of linear matrix equations is a competitive alternative to dealing with the large (vector) linear systems classically stemming from the aforementioned...

In this paper, we present an efficient algorithm for model reduction of discrete-time index-2 descriptor systems arising in context of linear time invariant (LTI) control and stability of descriptor systems. We propose a balanced truncation based model order reduction (MOR) strategy which consists of two stages. In the first stage, we reformulate the descriptor system into a generalized system...

The treatment of two-dimensional random walks in the quarter plane leads to Markov processes which involve semi-infinite matrices having Toeplitz or block Toeplitz structure plus a low-rank correction. In particular, finding the steady state probability distribution requires computing the minimal nonnegative solution of the quadratic matrix equation $A_1X^2+A_0X+A_{-1}=X$, where the...