Gerhard Freiling

Solution and asymptotic behavior of coupled Riccati equations in jump linear systems

A new necessary and sufficient condition for the existence of a positive semidefinite solution of coupled Riccati equations occurring in jump linear systems is derived. By verifying a Riccati inequality it is shown that such a solution exists; in addition two numerical algorithms are given to compute it. An example is given to illustrate the proposed method. >

On global existence of solutions to coupled matrix Riccati equations in closed-loop Nash games

Presents comparison and global existence results for solutions of coupled matrix Riccati differential equations appearing in closed loop Nash games and in mixed H/sub 2//H/sub /spl infin//-type problems. Convergence of solutions is established for the diagonal case, solutions of the corresponding algebraic equations are discussed using numerical examples.

On the solution of discrete-time Markovian jump linear quadratic control problems

Abstract A necessary and sufficient condition for the existence of a positive-semidefinite solution of the coupled algebraic discrete-time Riccati-like equation occurring in Markovian jump control problems is derived. By verifying a simple matrix inequality, it is shown that such a solution exists and can be obtained as a limit of a monotonic sequence. This leads to a straightforward numerical algorithm for the computation of the solution. An example is given to illustrate the proposed method.

A survey of nonsymmetric Riccati equations

Abstract We survey recent and also older results on nonsymmetric matrix Riccati differential equations and—in the time invariant case—on the corresponding algebraic Riccati equations. In particular we cite various applications connected with matrix Riccati equations.

Existence, Uniqueness, and Parametrization of Lagrangian Invariant Subspaces

The existence, uniqueness, and parametrization of Lagrangian invariant subspaces for Hamiltonian matrices is studied. Necessary and sufficient conditions and a complete parametrization are given. Some necessary and sufficient conditions for the existence of Hermitian solutions of algebraic Riccati equations follow as simple corollaries.

Inverse problems for Sturm–Liouville equations with boundary conditions polynomially dependent on the spectral parameter

Sturm–Liouville differential operators in a finite interval with boundary conditions depending polynomially on the spectral parameter are studied. We establish the properties of the spectral characteristics and investigate three inverse problems of recovering the operator either from the so-called Weyl function, or from discrete spectral data or from two spectra. For these inverse problems we provide procedures for constructing their solutions by the method of spectral mappings.

Existence and comparison theorems for algebraic Riccati equations and Riccati differential and difference equations

We present comparison and global existence theorems for solutions of generalized matrix Riccati differential and difference equations. Moreover we obtain existence and comparison results for the maximal solutions of the corresponding generalized algebraic Riccati equations. For the symplectic matrix Riccati differential equation we derive sufficient conditions ensuring the global existence of the solutions of the corresponding initial value problems.

On a class of rational matrix differential equations arising in stochastic control

We prove a monotonicity and a comparison theorem for the solutions of a rational matrix differential equation appearing in stochastic control and derive existence and convergence results for the solutions of this differential equation. Moreover, in the time-invariant case, we present conditions ensuring that the corresponding algebraic matrix equation has a stabilizing solution.

Properties of the solutions of rational matrix difference equations

Abstract We prove a comparison theorem for the solutions of a rational matrix difference equation, generalizing the Riccati difference equation, and existence and convergence results for the solutions of this equation. Moreover, we present conditions ensuring that the corresponding algebraic matrix equation has a stabilizing or almost stabilizing solution.

Necessary conditions for constant solutions of coupled Riccati equations in Nash games

Abstract The asymptotic behaviour of solutions as t → ∞ for coupled matrix Riccati equations occurring in open-loop linear-quadratic Nash games is studied in this paper. A general formula representing all possible solutions is given. Necessary conditions for constant real solutions are derived and an estimate for the rate of convergence is obtained. Two examples illustrate these results.