L^p optimal feedback control of homoclinic bifurcation in a forced Duffing oscillator

This paper proposes an L^p optimal control method to shift the homoclinic bifurcation of single-degree-of-freedom nonlinear oscillators. The homoclinic intersection between stable and unstable manifolds is detected analytically using the Melnikov method. An optimization strategy is formulated to obtain the control gain that minimizes the absolute value of the largest magnitude characteristic multiplier of the periodic orbit that emerges due to the shift of the homoclinic bifurcation. The optimal control gain is obtained by considering the stability of this periodic orbit. The cross-entropy method is used to obtain a numerical optimal solution to the optimization problem and, consequently, to obtain an optimal controller for the feedback method presented here. As an example of the L^p optimal control strategy, we consider the periodically forced Duffing oscillator with a twin-well potential. The numerical results demonstrate the capability of the L^p optimal control procedure to shift the homoclinic bifurcation. Moreover, the L^p optimal control can help regularize the fractal basin boundaries of the two confined attractors in the two potential wells.