The turnpike property and the longtime behavior of the Hamilton–Jacobi–Bellman equation for finite-dimensional LQ control problems

Abstract

We analyze the consequences that the so-called turnpike property has on the longtime behavior of the value function corresponding to a finite-dimensional linear-quadratic optimal control problem with general terminal cost and constrained controls. We prove that, when the time horizon T tends to infinity, the value function asymptotically behaves as $$W(x) + c\, T + \lambda $$ , and we provide a control interpretation of each of these three terms, making clear the link with the turnpike property. As a by-product, we obtain the longtime behavior of the solution to the associated Hamilton–Jacobi–Bellman equation in a case where the Hamiltonian is not coercive in the momentum variable. As a result of independent interest, we showed that linear-quadratic optimal control problems with constrained control enjoy a turnpike property, also particularly when the steady optimum may saturate the control constraints.