  We consider the problem of inverse optimal control design for systems that
are not affine in the control. In particular, we consider some classes of
partial differential equations (PDEs) with quadratic convection and
counter-convection, for which the L2 norm is a control Lyapunov function (CLF)
whose derivative has either a depressed cubic or a quadratic dependence in the
boundary control input. We also consider diffusive PDEs with or without linear
convection, for which a weighted L2 norm is a CLF whose derivative has a
quadratic dependence in the control input. For each structure on the derivative
of the CLF, we achieve inverse optimality with respect to a meaningful cost
functional. For the case where the derivative of the CLF has a depressed cubic
dependence in the control, we construct a cost functional for which the unique
minimizer is the unique real root of a cubic polynomial: the Cardano-Lyapunov
controller. When the derivative of the CLF is quadratic in the control, we
construct a cost functional that is minimized by two distinct feedback laws,
that correspond to the two distinct real roots of a quadratic equation. We show
how to switch from one root to the other to reduce the control effort.
