Optimal actuator design for linear systems with multiplicative noise

This paper addresses optimal actuator design for linear systems with process noise. It is well-known that the control that minimizes a quadratic cost in the state and control for a system with linear dynamics corrupted by additive Gaussian noise is of feedback type and its design depends on the solution of an associated Riccati equation. We consider here the case where the noise is multiplicative, by which we mean that its intensity is dependent on the state. We show how to derive the actuator that minimizes a linear quadratic cost. The solution requires to optimize a function defined on a manifold of low rank matrices; we provide a gradient descent algorithm to perform this optimization and show that this gradient descent converges to the global minimum almost surely.