Optimal control of fractional systems: numerics under diffusive formulation

Optimal control of fractional linear systems on a finite horizon can be classically formulated using the adjoint system. But the adjoint of a causal fractional integral or derivative operator happens to be an anti-causal operator: hence, the adjoint equations are not easy to solve in the first place. Using an equivalent diffusive realization helps transform the original problem into a coupled system of PDEs, for which the adjoint system can be more easily derived and properly studied. The numerical methods available to solve the LQR problem under equivalent diffusive formulation are investigated. The need for a discretization of the diffusive part that is low dimensional but still accurate proves crucial at this stage, especially to solve the dynamic Riccati equation. This is the reason why different methods will be presented and compared: either standard numerical quadrature, or optimal choice of the diffusive weights. The latter strategy proves more efficient, numerically speaking: L2 criterion and weighted-L2 will be recalled first with closed-form solutions, whereas L1 criteria will be also looked at, and equivalently reformulated into a linear programming problem, that can be efficiently solved thanks to the simplex algorithm. Numerical simulations will be provided to illustrate the practical feasability of this approach, mainly on the model of an oscillator damped by two types of memory variables, either as low-pass (fractional integral) or as high-pass (fractional derivative) filters: this is nothing but a toy model for the fully infinite-dimensional model of the Webster-Lokshin wave equation with somewhat idealized boundary control operators