  We propose the first generalization of Sontag s universal controller to
systems not affine in the control, particularly, to PDEs with boundary
actuation. We assume that the system admits a control Lyapunov function (CLF)
whose derivative, rather than being affine in the control, has either a
depressed cubic, quadratic, or depressed quartic dependence on the control. For
each case, a continuous universal controller that vanishes at the origin and
achieves global exponential stability is derived. We prove our result in the
context of convectionreaction-diffusion PDEs with Dirichlet actuation. We show
that if the convection has a certain structure, then the L2 norm of the state
is a CLF. In addition to generalizing Sontag s formula to some non-affine
systems, we present the first general Lyapunov approach for boundary control of
nonlinear PDEs. We illustrate our results via a numerical example.
