Abstract Two classes of recursive algorithms for computing the extreme solutions of the matrix equations X+AT X−1A = Q and X−AT X−1A = Q are presented. The Per Step Algorithms are based on the fixed point iteration method and its variations; the proposed Per Step Algorithm is an inversion free variant of the fixed point iteration method. The Doubling Algorithms are based on the cyclic reduction...

In this article, we investigate the simplified Modified Camassa-Holm (MCH) equation involving parameters by applying the (G'/G)-expansion method together with the generalized Riccati equation. In addition, G'( ) = c + aG( ) + bG ( ) is used, as an auxiliary equation and called generalized Riccati equation, where a, 2 b and c are arbitrary constants. The obtained traveling wave solutions includi...

Numerically reliable algorithms to compute the periodic non-negative definite stabilizing solutions of the periodic differential Riccati equation (PRDE) and discrete-time periodic Riccati equation (DPRE) are proposed. For the numerical solution of PRDEs, a new multiple shooting-type algorithm is developed to compute the periodic solutions in an arbitrary number of time moments within one period...

Group theoretical methods are used to study some properties of the Riccati equation, which is the only differential equation admitting a nonlinear superposition principle. The Wei–Norman method is applied for obtaining the associated differential equation in the group SL(2,R). The superposition principle for first order differential equation systems and Lie-Scheffers theorem are also analysed f...

In this paper a very general matrix quadratic equation is considered. This equation is known in literature as asymmetric Algebraic Riccati Equation (ARE) and arises in the solution of many problems in system and control theory and applied mathematics. For this reason a large amount of work has been produced on this topic. We present a new parametrization of the set of solutions of such equation...

If two solutions Y ≤ Z of the DARE are given then the set of solutions X with Y ≤ X ≤ Z can be parametrized by invariant subspaces of the closed loop matrix corresponding to Y . The paper extends the geometric theory of Willems from the continuous-time to the discrete-time ARE making the weakest possible assumptions.

We study a class of rational matrix differential equations that generalize the Riccati differential equations. The generalization involves replacing positive definite “weighting” matrices in the usual Riccati equations with either semidefinite or indefinite matrices that arise in linear quadratic control problems and differential games−both stochastic and deterministic. The purpose of this pape...

In this paper the authors develop a new algorithm to solve the standard discrete-time algebraic Riccati equation by using a skewHamiltonian transformation and the square-root method. The algorithm is structure-preserving and efficient because the Hamiltonian structure is fully exploited and only orthogonal transformations are used. The efficiency and stability of the algorithm are analyzed. Num...

A new backward stable, structure preserving method of complexity O(n) is presented for computing the stable invariant subspace of a real Hamiltonian matrix and the stabilizing solution of the continuous-time algebraic Riccati equation. The new method is based on the relationship between the invariant subspaces of the Hamiltonian matrix H and the extended matrix [ 0 H H 0 ] and makes use of the ...