Optimal Steering of a Linear Stochastic System to a Final Probability Distribution, Part I


We consider the matter of steering a linear dynamical system with complete state observation from an initial Gaussian distribution in state-space to a final one with minimum energy management. The system is stochastically driven through the management channels; an example for such a system is that of an inertial particle experiencing random “white noise” forcing. We tend to show that a target chance distribution can continually be achieved in finite time. The optimal control is given in state-feedback form and is computed explicitly by solving a try of differential Lyapunov equations that are nonlinearly coupled through their boundary values. This result, given its enticing algorithmic nature, appears to own many potential applications like to quality management, control of business processes, likewise on active control of nanomechanical systems and molecular cooling. The matter to steer a diffusion process between finish-point marginals includes a long history (Schrödinger bridges) and the current case of steering a linear stochastic system constitutes such a Schrödinger bridge for possibly degenerate diffusions. Our results offer the first implementable kind of the optimal management for a general Gauss-Markov process. Illustrative examples are provided for steering inertial particles and for “cooling” a stochastic oscillator. A upshot establishes directly the property of Schrödinger bridges as the foremost seemingly random evolution between given marginals to the present context of linear stochastic systems. A second half to this work, that is to appear as part II, addresses the general scenario where the stochastic excitation enters through channels that will differ from those used to manage.

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