On the Application of Optimal Control Techniques to the Shadowing Approach for Time Averaged Sensitivity Analysis of Chaotic Systems

Traditional sensitivity analysis methods fail for chaotic systems due to the unstable characteristics of the linearized equations. To overcome these issues two methods have been developed in the literature, one being the shadowing approach, which results in a minimization problem, and the other being numerical viscosity, where a damping term is added to the linearized equations to suppress the instability. The shadowing approach is computationally expensive but produces accurate sensitivities, while numerical viscosity can produce less accurate sensitivities but with significantly reduced computational cost. However, it is not fully clear how the solutions generated by these two approaches compare to each other. In this work we aim to bridge this gap by introducing a control term, found with optimal control theory techniques, to prevent the exponential growth of solution of the linearized equations. We will refer to this method as optimal control shadowing. We investigate the computational aspects and performance of this new method on the Lorenz and Kuramoto-Sivashinsky systems and compare its performance with simple numerical viscosity schemes. We show that the tangent solution generated by the proposed approach is similar to that generated by shadowing methods, suggesting that optimal control attempts to stabilize the unstable shadowing direction. Further, for the spatially extended system, we examine the energy budget of the tangent equation and show that the control term found via the solution of the optimal control problem acts only at length scales where production of tangent energy dominates dissipation, which is not necessarily the case for the numerical viscosity methods.