The exponential turnpike property for periodic linear quadratic optimal control problems in infinite dimension

In this paper, we establish an exponential periodic turnpike property for linear quadratic optimal control problems governed by periodic systems in infinite dimension. We show that the optimal trajectory converges exponentially to a periodic orbit when the time horizon tends to infinity. Similar results are obtained for the optimal control and adjoint state. Our proof is based on the large time behavior of solutions of operator differential Riccati equations with periodic coefficients.