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We consider the problem to steer a linear dynamical system with full stateobservation from an initial gaussian distribution in state-space to a final onewith minimum energy control. The system is stochastically driven through thecontrol channels; an example for such a system is that of an inertial particleexperiencing random "white noise" forcing. We show that a target probabilitydistribution can always be achieved in finite time. The optimal control isgiven in state-feedback form and is computed explicitely by solving a pair ofdifferential Lyapunov equations that are coupled through their boundary values.This result, given its attractive algorithmic nature, appears to have severalpotential applications such as to active control of nanomechanical systems andmolecular cooling. The problem to steer a diffusion process between end-pointmarginals has a long history (Schr\"odinger bridges) and therefore, the presentcase of steering a linear stochastic system constitutes a Schr\"odinger bridgefor possibly degenerate diffusions. Our results, however, provide the firstimplementable form of the optimal control for a general Gauss-Markov process.Illustrative examples of the optimal evolution and control for inertialparticles and a stochastic oscillator are provided. A final result establishesdirectly the property of Schr\"{o}dinger bridges as the most likely randomevolution between given marginals to the present context of linear stochasticsystems.

Optimal steering of a linear stochastic system to a final probability distribution