  The paper develops an optimal regulator for a general class of multi-input
affine nonlinear systems minimizing a nonlinear cost functional with infinite
horizon. The cost functional is general enough to enforce saturation limits on
the control input if desired. An efficient algorithm utilizing tensor algebra
is employed to compute the tensor coefficients of the Taylor series expansion
of the value function (i.e., optimal cost-to-go). The tensor coefficients are
found by solving a set of nonlinear matrix equations recursively generalizing
the well-known linear quadratic solution. The resulting solution generates the
optimal controller as a nonlinear function of the state vector up to a
prescribed truncation order. Moreover, a complete convergence of the computed
solution together with an estimation of its applicability domain are provided
to further guide the user. The algorithm's computational complexity is shown to
grow only polynomially with respect to the series order. Finally, several
nonlinear examples including some with input saturation are presented to
demonstrate the efficacy of the algorithm to generate high order Taylor series
solution of the optimal controller.
