Indefinite mixed H₂/H∞ control of linear stochastic systems

We introduce an indefinite generalisation to the finite-horizon mixed $\mathrm{H}_{2} / \mathrm{H}_{\infty}$ control method for linear stochastic systems with additive and multiplicative noise. This permits for the consideration of linear systems without feed-through input to output paths, and optimality criteria with indefinite weights. We prove that in this case there exist a parameterised family of Nash equilibria of an affine state-feedback form, and derive explicit formulas for such equilibria in terms of certain coupled Riccati and linear differential equations with equality and inequality algebraic constraints.