The dual algebraic Riccati equations and the set of all solutions of the discrete-time Riccati equation

ABSTRACT

In this paper, two new pairs of dual continuous-time algebraic Riccati equations (CAREs) and dual discrete-time algebraic Riccati equations (DAREs) are proposed. The dual DAREs are first studied with some nonsingularity assumptions on the system matrix and the parameter matrix. Then, in the case of singular matrices, a generalised inverse is introduced to deal with the dual DARE problem. These dual AREs can easily lead us to an iterative procedure for finding the anti-stabilising solutions, especially to DARE, by means of that for the stabilising solutions. Furthermore, we provide the counterpart results on the set of all solutions to DARE inspired by the results for CARE. Two examples are presented to illustrate the theoretical results.

KEYWORDS:

• Riccati equation
• anti-stabilising solution
• solution set
• invariant subspace
• Cayley transform

Acknowledgements

The authors would like to thank the editor and the anonymous reviewers for their insightful suggestions. This work was supported by the National Natural Science Foundation of China under Grant 61374053 and Grant 61422308.

Disclosure statement

No potential conflict of interest was reported by the authors.

Notes

1. See Damm (2004)(Chapter 5) for details.

Additional information

Funding

National Natural Science Foundation of China [grant numbers 61374053 and 61422308].