For continuous-time systems, the optimal gain matrix is

, where

is the solution of the continuous-time Riccati equation:

For discrete-time systems, the optimal gain matrix is

, where

is the solution of the discrete-time Riccati equation:

For an LQR design, the Nyquist plot of the open-loop transfer function always lies outside the unit circle centered at

:

Consequently, the gain margin

and the phase margin

satisfies

and

:

The gains associated with a state increase as its weight is increased:

The higher penalty of the second state reduces its overshoot:

For stable open-loop systems, the gain

converges to zero as

approaches zero or

approaches infinity:

As the gain decreases, the closed-loop poles move closer to the open-loop ones:

For unstable open-loop systems, the gain

converges to a minimum as

approaches zero or

approaches infinity:

As the gain decreases, the closed-loop poles move closer to any stable open-loop poles and the stable mirror images of the unstable ones:

When

approaches infinity or

approaches zero, the gain

becomes unbounded:

As the gains increase, the states are penalized more and their values become smaller:

The optimal cost-to-go is a Lyapunov function:

The state trajectory projected on the optimal cost surface asymptotically approaches the origin:

The optimal state trajectory for a system with one state:

The co-state trajectory:

The optimal input trajectory:

The optimal cost trajectory:

The optimal cost satisfies the infinite horizon Hamilton-Jacobi-Bellman equation

, where

:

The optimal input minimizes the Hamiltonian, thus satisfying

: