## Abstract

Differential Riccati equations are at the heart of many applications in control theory. They are time-dependent, matrix-valued, and in particular nonlinear equations that require special methods for their solution. Low-rank methods have been used heavily for computing a low-rank solution at every step of a time-discretization. We propose the use of an all-at-once space-time solution leading to a large nonlinear space-time problem for which we propose the use of a Newton–Kleinman iteration. Approximating the space-time problem in a higher-dimensional low-rank tensor form requires fewer degrees of freedom in the solution and in the operator, and gives a faster numerical method. Numerical experiments demonstrate a storage reduction of up to a factor of 100.

Original language | English |
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Pages (from-to) | 407-429 |

Number of pages | 23 |

Journal | Numerical Algebra, Control and Optimization |

Volume | 11 |

Issue number | 3 |

DOIs | |

Publication status | Published - 30 Sep 2021 |

## Keywords

- Low-rank methods
- Non-linear problems
- Optimal Control
- Riccati equations

## ASJC Scopus subject areas

- Algebra and Number Theory
- Control and Optimization
- Applied Mathematics