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The LQR problem for a time-periodic system of the form

\begin{align} \dot{x} = A(t) x + B(t) u, \quad t \in [0, \infty), \quad x(0) = x_i \label{eq:time-varying-system}\\ A(t+T) = A(t),\ B(t + T) = B(t) \nonumber \end{align}

is as follows. With a quadratic form defined on \( (x,u) \) pairs

\begin{align} \label{eq:quadratic-form} \mathbf{q}(x, u) := \lim_{t_f \to \infty} \int_{0}^{t_f} \begin{bmatrix} x \\ u \end{bmatrix}^{\star} \begin{bmatrix} Q & 0 \\ 0 & r \end{bmatrix} \begin{bmatrix} x \\ u \end{bmatrix} =: \lim_{t_f \to \infty} \int_{0}^{t_f} q(x,u) dt \end{align}

with \( q \ge 0 \) and \( r \ge 0 \), find the infimum of the quadratic form \( \mathbf{q} \) subject to the dynamics: \[ \inf_{x,u} \mathbf{q}(x,u). \]

\begin{align} \label{eq:lqr-inf-via-duality} \inf_{x, u} \mathbf{q}(x, u) = x_i^{\star} \bar{\lambda}(0) x_i, \end{align}

where \( \bar{\lambda} \) is the maximal solution of the differential linear matrix inequality over \( [0, t] \).