Abstract
We consider the Linear-Quadratic-Regulator (LQR) problem in terms of optimizing a real-valued matrix function over the set of feedback gains. Such a setup facilitates examining the implications of a natural initial-state independent formulation of LQR in designing first order algorithms. We characterize several analytical properties (smoothness, coerciveness, quadratic growth) that are crucial in the analysis of gradient-based algorithms. We then examine three types of well-posed flows for LQR: gradient flow, natural gradient flow and the quasi-Newton flow. The coercive property suggests that these flows admit unique solutions while gradient dominated property indicates that the corresponding Lyapunov functionals decay at an exponential rate; quadratic growth on the other hand guarantees that the trajectories of these flows are exponentially stable in the sense of Lyapunov.
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