The problem of using an adaptive optics system to correct for nonlinear effects like thermal blooming is addressed using a model containing nonlinear lenses through which Gaussian beams are propagated. The best correction of this nonlinear system can be formulated as a deterministic open loop optimal control problem. This treatment gives us a limit for the best possible correction. Aspects of adaptive control and servo systems are not included at this stage. We ask for that control in the transmitter plane which minimizes the time averaged area or maximizes the fluence in the target plane. The standard minimization procedure leads to a two-point-boundary-value problem, which is ill-conditioned in our case. We were able to solve the optimal control problem using an iterative gradient technique. An instantaneous correction is introduced and compared with the optimal correction. The results of our calculations show that for short times or weak nonlinearities the instantaneous correction is close to the optimal correction, but that for long times and strong nonlinearities a large difference develops between the two types of correction. For these cases the steady state correction becomes better than the instantaneous correction and approaches the optimum correction.
Jan Herrmann, Jan Herrmann,
"Optimal Control In Adaptive Optics Modeling Of Nonlinear Systems", Proc. SPIE 0365, Adaptive Optics Systems and Technology, (23 August 1983); doi: 10.1117/12.934202; https://doi.org/10.1117/12.934202