An exponential turnpike theorem for dissipative discrete time optimal control problems

revised 2013, 23 p.International audienceWe investigate the exponential turnpike property for nite horizon undercounted discrete time optimal control problems without any terminal constraints. Considering a class of strictly dissipative systems we derive a boundedness condition for an auxiliary optimal value function which implies the exponential turnpike property. Two theorems illustrate how this boundedness condition can be concluded from structural properties like controllability and stabilizability of the control system under consideration

Turnpike in infinite dimension

Let $\Phi$ be a correspondence from a normed vector space $X$ into itself, let $u: X\to \mathbf{R}$ be a function, and $\mathcal{I}$ be an ideal on $\mathbf{N}$. Also, assume that the restriction of $u$ on the fixed points of $\Phi$ has a unique maximizer $\eta^\star$. Then, we consider feasible paths $(x_0,x_1,\ldots)$ with values in $X$ such that $x_{n+1} \in \Phi(x_n)$ for all $n\ge 0$. Under certain additional conditions, we prove the following turnpike result: every feasible path $(x_0,x_1,\ldots)$ which maximizes the smallest $\mathcal{I}$-cluster point of the sequence $(u(x_0),u(x_1),\ldots)$ is necessarily $\mathcal{I}$-convergent to $\eta^\star$. We provide examples that, on the one hand, justify the hypotheses of our result and, on the other hand, prove that we are including new cases which were previously not considered in the related literature.Comment: Example 2.6 has been adde

Turnpike in optimal shape design

We introduce and study the turnpike property for time-varying shapes, within the viewpoint of optimal control. We focus here on second-order linear parabolic equations where the shape acts as a source term and we seek the optimal time-varying shape that minimizes a quadratic criterion. We first establish existence of optimal solutions under some appropriate sufficient conditions. We then provide necessary conditions for optimality in terms of adjoint equations and, using the concept of strict dissipativity, we prove that state and adjoint satisfy the measure-turnpike property, meaning that the extremal time-varying solution remains essentially close to the optimal solution of an associated static problem. We show that the optimal shape enjoys the exponential turnpike property in term of Hausdorff distance for a Mayer quadratic cost. We illustrate the turnpike phenomenon in optimal shape design with several numerical simulations