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

On the asymptotic behavior of the value function in large time optimal control problems

“…In this manuscript, we extend the results of [18,19] to the class of dissipative nonlinear systems. The concept of (strict) dissipativity, as introduced in [21], is defined in a broad context, accompanied by related notions like the available storage function and the supply rate function.…”

“…Theorem 5.1. Under Assumptions pA 1 q´pA 8 q, the value function (4) satisfies vpT, x, zq " T.v `vf pxq `vb pzq `op1q (18) as T Ñ `8.…”

“…Considering a general nonlinear dissipative optimal control problem in finite dimension and fixed time T , we have established a two-term asymptotic expansion of the value function as T Ñ `8. Essentially based on the strict dissipativity property, the result is a generalization of the expansion established in the variants of the linear quadratic case treated in [18] and [19]. We highlight here that a generalization of [19] to the infinite dimensional case is currently being finalized.…”

“…Figure 2: Turnpike phenomenon Significant insights have been achieved in [18] and [19], where the authors explore variations of the linear quadratic (LQ) problem. Assuming certain controllability conditions, they derive the large-time expansion of the value function at the order of one in 1{T .…”

“…Assuming certain controllability conditions, they derive the large-time expansion of the value function at the order of one in 1{T . In [18], the various aspects are identified, and some are interpreted within the framework of the Hamilton-Jacobi-Bellman (HJB) theory. Meanwhile, in [19], all the terms are characterized as functions of the solutions of the Riccati algebraic equation, initial/final states, and the Lagrange multiplier of the static optimization problem.…”