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Linear-Quadratic Problems in Systems and Controls via Covariance Representations and Linear-Conic Duality: Finite-Horizon Case

Linear-Quadratic (LQ) problems that arise in systems and controls include the classical optimal control problems of the Linear Quadratic Regulator (LQR) in both its deterministic and stochastic forms, as well as $H^\infty$-analysis (the Bounded Real Lemma), the Positive Real Lemma, and general Integral Quadratic Constraints (IQCs) tests. We present a unified treatment of all of these problems using an approach which converts linear-quadratic problems to matrix-valued linear-linear problems with a positivity constraint. This is done through a system representation where the joint state/input covariance (the outer product in the deterministic case) matrix is the fundamental object. LQ problems then become infinite-dimensional semidefinite programs, and the key tool used is that of linear-conic duality. Linear Matrix Inequalities (LMIs) emerge naturally as conal constraints on dual problems. Riccati equations characterize extrema of these special LMIs, and therefore provide solutions to the dual problems. The state-feedback structure of all optimal signals in these problems emerge out of alignment (complementary slackness) conditions between primal and dual problems. Perhaps the new insight gained from this approach is that first LMIs, and then second, Riccati equations arise naturally in dual, rather than primal problems. Furthermore, while traditional LQ problems are set up in $L^2$ spaces of signals, their equivalent covariance-representation problems are most naturally set up in $L^1$ spaces of matrix-valued signals.