Question: How to solve continuous time LQR optimal control problem?

Suppose we have a "Finite-horizon, continuous-time LQR" optimal control problem. I am wondering how to solve Riccati DE and then determine state and control over time. Is it possible to analytically solve Riccati DEs? Suppose we have this example and we want to solve the problem based on the formulas in Wikipedia webpage.

A := Matrix(2, 2, [[2, 1], [1, -1]])

B := Vector[column](2, [3, 1])

Q := Matrix(2, 2, [[3, 0], [0, 2]])

R := 3

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