A Periodic Dichotomy in Linear Control Theory

Abstract

In this paper, we construct a periodic dichotomy transformation using solutions of periodic Riccati and Lyapunov equations. As an application of this transformation, we provide an explicit representation of the optimal extremal for periodic linear quadratic optimal control problems. Specifically, we establish a complete characterization of the optimal extremal under suitable exponential stabilizability and detectability assumptions.

Figures (6)

• Figure 1: Two distinct periodic orbits of $P_{11}$
• Figure 2: Characteristic multipliers (eigenvalues) of $P$
• Figure 3: Matrix $P(\cdot)$
• Figure 4: Matrix $E(\cdot)$
• Figure 5: Periodic optimal extremal $(y_\theta, \lambda_\theta, u_\theta)$

Theorems & Definitions (7)

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• proof : Proof of Theorem \ref{['thm:analy_riccati']}