  In this paper, we consider the problem of multi-objective optimal control of
a dynamical system with additive and multiplicative noises with given second
moments and arbitrary probability distributions. The objectives are given by
quadratic constraints in the state and controller, where the quadratic forms
maybe indefinite and thus not necessarily convex. We show that the problem can
be transformed to a semidefinite program and hence convex. The optimization
problem is to be optimized with respect to a certain variable serving as the
covariance matrix of the state and the controller. We show that affine
controllers are optimal and depend on the optimal covariance matrix.
Furthermore, we show that optimal controllers are linear if all the quadratic
forms are convex in the control variable. The solutions are presented for both
the finite and infinite horizon cases. We give a necessary and sufficient
condition for mean square stabilizability of the dynamical system with additive
and multiplicative noises. The condition is a Lyapunov-like condition whose
solution is again given by the covariance matrix of the state and the control
variable. The results are illustrated with an example.
