---
tag: Y4V154JWR7
kind: concept
name: Riccati differential equation of optimal control
msc: [49N10, 34A34, 93C05]
sources:
  - url: https://en.wikipedia.org/wiki/Riccati_equation
---

## Summary

The Riccati differential equation of optimal control is a quadratic matrix differential equation for a symmetric matrix function $P(t)$.
It is solved backward in time from a terminal condition.
Its solution gives the optimal feedback law and the optimal cost of the [[linear-quadratic regulator|HR720TJZZR]].
It is the matrix analogue of the scalar [[Riccati equation|0VT2QK6X6Q]], and its steady states solve the [[algebraic Riccati equation|GGZV75ZZMV]].

## Explanations

### L1 human

For a linear system with a quadratic cost, the optimal remaining cost from state $x$ at time $t$ is a quadratic form $x^T P(t) x$.
The matrix $P(t)$ answers the question "how expensive is it to be at $x$ at time $t$?".
The Riccati equation tells how this matrix changes as one moves backward from the final time.
Once $P(t)$ is known, the best control is a simple linear function of the current state.