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

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