Motivation I have been interested in understanding the relationships between some frequency domain conditions, Riccati equations and matrix inequalities (such as the question asked here). I came to realize that this paper
J. Willems, "Least squares stationary optimal control and the algebraic Riccati equation," in IEEE Transactions on Automatic Control, vol. 16, no. 6, pp. 621-634, December 1971, doi: 10.1109/TAC.1971.1099831
is an important resource in this regard, as also identified in the accepted answer here. As I am studying this paper, I have run into several questions related to definitions and results in Section 3. I will restate them as I understand them and ask my questions.
Relevant function spaces
- $L_{2e}^+ = \{f:[0,\infty) \rightarrow \mathbb{R}^p | f \in L_2(0,T)\ \text{for all}\ T\ge 0 \}$
- $L_{2e}^- = \{f:(-\infty,0] \rightarrow \mathbb{R}^p | f \in L_2(-T,0)\ \text{for all}\ T\ge 0 \}$
My question about the significance of such extended $L_2$ spaces was answered here.
Linear dynamics All the minimization problems below are constrained by the linear dynamics, $\dot{x} = A x + B u$, and the initial condition, $x(0) = x_0$, where $(A,B)$ is controllable.
Minimization problems Let $w$ be a quadratic form on $\left(\begin{smallmatrix}x \\ u\end{smallmatrix} \right)$, not necessarily positive definite. Four minimization problems are defined in Section 3 of the Willems paper, all constrained by the above linear dynamics.
- $\begin{equation} V_f^+(x_0) = \inf\limits_{u\in L_{2e}^+} \int_0^\infty w(x,u) dt \end{equation}$
- $\begin{equation} V^+(x_0) = \inf\limits_{u\in L_{2e}^+} \int_0^\infty w(x,u) dt\quad \text{subject to}\ \lim\limits_{t\rightarrow\infty} x(t) = 0 \end{equation}$
- $\begin{equation} V^-(x_0) = -\inf\limits_{u\in L_{2e}^-} \int_0^\infty w(x,u) dt\quad \text{subject to}\ \lim\limits_{t\rightarrow-\infty} x(t) = 0 \end{equation}$
- $\begin{equation} V_n^+(x_0) = \inf\limits_{\substack{u\in L_{2e}^+ \\ T\ge0}} \int_0^T w(x,u) dt \end{equation}$
Question 1 I am not sure if my following interpretation of minimization problem 3 is correct. The dynamics is integrated backwards in time, so stated more explicitly, the problem is $$\begin{equation} \begin{gathered} \sup\limits_{u\in L_{2e}^-} \int_0^{-\infty} w(x(t),u(t)) dt \\ \begin{aligned} \text{subject to:} & -\dot{x}(t) = A x(t) + B u(t)\\ & x(0) = x_0 \\ & \lim\limits_{t\rightarrow-\infty} x(t) = 0 \end{aligned} \end{gathered} \end{equation}$$ (where the minus sign on the left hand side of the differential equation is because of integrating backwards). To turn this into the form in the paper, substitute $t=-\tau$ in the integral to be maximized. $$ \begin{equation} \begin{gathered} \sup\limits_{u\in L_{2e}^-} \int_0^{-\infty} w(x(t),u(t)) dt = \sup\limits_{u\in L_{2e}^-} -\int_0^\infty w(x(-\tau),u(-\tau)) d\tau = -\inf\limits_{u\in L_{2e}^-} \int_0^\infty w(x(-\tau),u(-\tau)) d\tau \\ = -\inf\limits_{u\in L_{2e}^-} \int_0^\infty w(x(-t),u(-t)) dt \end{gathered} \end{equation} $$ where in the last step, $\tau$ has been renamed to $t$ for convenience. Thus my interpretation of minimization problem 3 is $$\begin{equation} \begin{gathered} V^-(x_0) = -\inf\limits_{u\in L_{2e}^-} \int_0^\infty w(x(-t),u(-t)) dt \\ \begin{aligned} \text{subject to:} & -\dot{x}(t) = A x(t) + B u(t)\\ & x(0) = x_0 \\ & \lim\limits_{t\rightarrow-\infty} x(t) = 0 \end{aligned} \end{gathered} \end{equation}$$
- Is this what is intended in the definition?
- If so, why is the functional being maximized when going backwards in time and minimized when going forwards?
Question 2 There is a statement in the paper, "By controllability, $V^-$ is bounded below". Why is this true?
My attempt at a proof: By controllability, for any $T<0$, there is a bounded $u(\cdot)$ such that $x(T) = 0$, and the quadratic form evaluated on such a trajectory and integrated is finite. I am not sure how to prove that the $\sup$ over all such $u(\cdot)$s is bounded.
Question 3 A couple of results that arise in the statement and proof of Theorem 1 in Section 3: If for a quadratic form $w$, $\int_0^T w(x,u) dt \ge 0$ for every $T\ge 0$ and every trajectory $(x(\cdot),u(\cdot))$ constrained by $\dot{x} = A x + B u$ (with (A,B) controllable) and $x(0) = 0$, then
- $V^-(x_0) \le 0$ for any $x_0$
- $-\int_{-\infty}^0 w(x,u) dt \le \int_0^T w(x,u) dt$
I am not able to even think about how to prove these, since in the hypothesis of this statement, the integration is over positive $t$, so $u \in L_{2e}^+$, whereas the results of the statement involve $V^-$ and an integral over negative $t$, so $u \in L_{2e}^-$. I am wondering if my interpretation of the statement is correct. (The paper states that this statement is obvious).
