Stabilizing unstable periodic orbits of chaotic systems via an optimal principle (Q5952981)

The authors propose an optimal control method for stabilizing unstable periodic orbits of chaotic ordinary differential equations by means of a delayed additive feedback control with the delay equal to the period of the orbit. NEWLINENEWLINENEWLINEUsing optimal control techniques, an explicit expression for the optimal controller minimizing the integral for one period over a weighted sum of quadratic expressions involving the trajectory and its first two derivatives is obtained. Under a stability condition on the weighting matrices, it is shown that the optimal cost tends to 0 as \(t\to\infty\). After some comments on possible extensions of the method, simulation results for the Rössler system are given in order to illustrate the performance.