# Any stabilizing control law $K$ is optimal for some LQR problems ($Q$ and $R$).

I recently went through Kalman's paper "When is a Linear Control System Optimal" published in 1964. The paper makes me wonder whether the following statement is true: Any stabilizing control law is optimal for some LQR problems.

For the linear system $$x_{t+1} = Ax_t + Bu_t,$$ and the average of a quadratic cost function $$$$J(\pi) = \lim_{T\rightarrow\infty} \sum_{t=0}^{T-1}\left(x_t'Qx_t + u_t'Ru_t \right),$$$$ where $$Q$$ is positive semi-definite and $$R$$ is positive definite, $$(A,B)$$ is controllable, and $$(Q^{1/2},A)$$ is observable, we know the policy $$K$$ such that $$\pi(x) = K x$$ is optimal for the cost $$J(\pi)$$ when $$K = -(R+B'PB)^{-1}B'PA, \tag{a}$$ where $$P$$ is the unique positive definite solution of the algebraic Riccati equation $$P = Q + A'PA- A'PB(R+B'PB)^{-1}B'PA \tag{b}.$$ We say $$K$$ is optimal for the LQR problem defined by $$Q\succeq 0$$ and $$R\succ 0$$ if conditions (a) and (b) are satisfied.

Or equivalently, we can say $$K$$ is optimal for $$Q$$ and $$R$$ if \begin{aligned} Q + A'P(A+BK) - P =0,\\ RK+ B'P(A+BK)=0,\\ P\succeq0,\ \ Q\succeq 0, R\succ0, \end{aligned} \tag{c} are satisfied.

The statement is that for any stabilizing $$K$$, there always exists some $$Q\succeq 0$$ and $$R\succ 0$$ such that conditions (a) and (b) or conditions (c) are satisfied (Any stabilizing control law is optimal for some LQR problems.)

I want to show that the statement is true if $$B\in\mathbb{R}^{n\times m}$$ with $$m has rank $$m$$.

My effort:

I tried to show the statement is false. I select some stabilizing $$K$$. But for every $$K$$ I tried, I can always find $$Q$$ and $$R$$ that make $$K$$ optimal.

A similar statement can be found in Theorem 7 of Kalman's paper. But in the original paper, Kalman adopted a different form of the cost function (i.e., the control is scalar and an additional cost term $$x_t'r u_t$$ is considered).

• I believe your question is equivalent to this question. Feb 19, 2022 at 6:01
• The discrete-time has been considered more recently and the results and conclusions may differ from those in the continuous-time case. You may look at some of the papers there: scholar.google.com/….
– KBS
Feb 19, 2022 at 12:43
• @KwinvanderVeen Thanks for the reference. From their post, the conclusion is every $K$ (not necessarily stabilizing) is optimal to a quadratic performance index that includes a cross-product term between the state and control. We don't have the cross-product term. Then several sufficient and necessary conditions need to be satisfied for a stabilizing $K$ to be optimal for some $Q$. These conditions are known in Theorem 1 of ieeexplore.ieee.org/abstract/document/6880317 and the discussion after that. Feb 19, 2022 at 22:12
• @KBS thanks for sharing. I checked Prof. HU's papers before I posted the question. No answer to this qustion was spotted from his recent papers about inverse optimal control. Feb 19, 2022 at 22:41

The statement that any stabilizing control law is optimal for some LQR problems is not necessarily true even when $$B$$ has rank $$m$$. The necessary and sufficient conditions for a stabilizing $$K$$ to be optimal can be found in Theorem 4.1 and Theorem 5.1 of [R1].
[R1] also gives an example in which $$K$$ is stabilizing under $$(A,B)$$ but not optimal for any given $$Q\succeq 0$$ (see Section 6 of [R1]).