Converse linear quadratic optimal control

Question

It is well known that for a linear time invariant system

$$ \dot{x} = A x + B u \tag{1} $$

with $(A, B)$ controllable, there exists a static state feedback $u = -K x$ such that the cost function

$$ J = \int_0^{\infty} x^T Q x + u^T R u \, dt \tag{2} $$

is minimized, assuming $Q \geq 0$ (positive semi-definite) and $R > 0$ (positive definite). The gain $K$ is the solution of the algebraic Riccati equation:

$$ \begin{align} 0 &= A^T P + P A - P B R^{-1} B^T P + Q \\ K &= R^{-1} B^T P \\ P &= P^T \geq 0 \end{align} $$

known as linear quadratic regulator (LQR). However, I wonder whether the converse also holds?

That is, given a stabilizing $K_s$ (such that $A - B K_s$ is Hurwitz), do there exist matrices $Q \geq 0$ and $R > 0$ such that $u = -K_s x$ minimizes $(2)$ given $(1)$? Or put differently:

Question: Is every stabilizing linear state feedback optimal in some sense?

Answer 1

See the paper: Kalman, R. E. (1964). When is a linear control system optimal?. Journal of Basic Engineering, 86(1), 51-60.

The answer is positive at least for a class of systems. As far as I remember, the answer is also positive for a general LTI system, but I cannot find a reference at the moment.

UPDATE: Every linear system with nondynamic feedback is optimal with respect to a quadratic performance index that includes a cross-product term between the state and control, see [R1].

If you do not allow for the cross-product term, then several sufficient and necessary conditions are known, see for example [R2] and the references there.

[R1] Kreindler, E., & Jameson, A. (1972). Optimality of linear control systems. IEEE Transactions on Automatic Control, 17(3), 349-351.

[R2] Priess, M. C., Conway, R., Choi, J., Popovich, J. M., & Radcliffe, C. (2015). Solutions to the inverse lqr problem with application to biological systems analysis. IEEE Transactions on control systems technology, 23(2), 770-777.

Answer 2

The problem is called the Inverse Problem of Optimal Control (see page 147 - 148).

Given a system $$\dot{x}=Ax+Bu,\qquad x(t_0)=x_0$$ $$ z = \begin{bmatrix} Q^{1/2}& 0 \\ 0 & R^{1/2}\\ \end{bmatrix}\begin{bmatrix}x\\u \end{bmatrix},$$

with $(A,B)$ is stabilizable, $(Q,A)$ is detectable and $R>0$ (positive definite). The linear quadratic regulator problem is given by minimizing

$$\int_{0}^{\infty}z^Tz dt.$$

From Boyd et al 1994: The inverse problem of optimal control is the following. Given a matrix $K$, determine if there exists $Q ≥ 0$ and $R > 0$, such that $(Q, A)$ is detectable and $u = Kx$ is the optimal control for the corresponding LQR problem. Equivalently, we seek $R > 0$ and $Q ≥ 0$ such that there exists P nonnegative and P1 positive-definite satisfying

$$(A + BK)^T P + P(A + BK) + K^TRK + Q = 0, \quad B^T P + RK = 0$$

and $A^T P_1 + P_1A < Q$. This is an LMIP in $P$, $P_1$, $R$ and $Q$. (The condition involving $P_1$ is equivalent to $(Q, A)$ being detectable.)