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This paper develops state-feedback and output-feedback control design methodologies for boundary stabilization of a class of systems that involves cascade connection of a nonlinear viscous Hamilton-Jacobi partial differential equation (PDE) and a possibly unstable linear ordinary differential equation (ODE). First, an explicit state-feedback controller is designed based on the infinite-dimensional backstepping method and by utilizing a locally invertible feedback linearizing transformation whose role is to convert the non-linear viscous Hamilton-Jacobi PDE to a linear heat equation. Next, an output-feedback controller is proposed which makes the closed-loop system exponentially stable through the measured output of the system. The main feature of the proposed output-feedback scheme is that the distributed integral terms are avoided in the feedback law. This is achieved by exploiting the derived state-feedback controller and introducing a virtual ODE-PDE system, whose ODE state determines the stabilizing feedback law. Then, an observer is proposed that generates state estimates of the virtual system. In both schemes, local exponential stability is shown via Lyapunov analysis and an estimate of the region of attraction is provided. Finally, simulation examples for an unstable ODE are presented to validate the effectiveness of the proposed results.This is an open access article under the terms of the Creative Commons Attribution License, which permits use, distribution and reproduction in any medium, provided the original work is properly cited.

The main purpose of the paper is to optimize state feedback parameters using intelligent method, GA, Hermite-Biehler, and chaos algorithm. GA is implemented for local search but it has some deficiencies such as trapping into a local minimum and slow convergence, so the combination of Hermite-Biehler and chaos algorithm has been added to GA to avoid its deficiencies. Dividing search space is usually done by distributed population genetic algorithm (DPGA). Moreover, using generalized Hermite-Biehler Theorem can find the domain of parameters. In order to speed up the convergence at the first step, Hermite-Biehler method finds some intervals for controller, in the next step the GA will be added, and, finally, chaos disturbance will help the algorithm to reach a global minimum. Therefore, the proposed method can optimize the parameters of the state feedback controller.