Introduction to the Asymptotic Analysis of Optimal Control Problems: A Parade of Examples

Abstract

The main interest in this book is in mathematical models of optimal control problems (OCPs) that depend on some small parameter ε. In many mathematical problems, which come from natural or engineering sciences, industrial applications, or abstract mathematical questions all by themselves, some parameters appear (small or large, of geometric or constitutive origin, coming from approximation processes or discretization arguments) and make those control problems increasingly complex or degenerate. For instance, if we deal with OCPs on reticulated structures or perforated domains, this small parameter comes from the geometry of such domains. This dependence can be defined both by a regular and a singular occurrence of the parameter ε (the presence of ε-periodic coefficients, ε-periodically perforated zone of controls, quickly oscillating boundaries, etc.). In order to proceed with a mathematical description, we formulate the following parameterized OCP (OCP _ε ), where ε is a small parameter:

$$(\mathrm{OCP}_{\varepsilon}):\qquad \min\left\{I_{\varepsilon}(u,y)\,:\ (u,y)\in \Xi_{\varepsilon}\right\},$$

where

(B ₁):

$I_{{\varepsilon}}:\mathbb{U}_{{\varepsilon}}\times \mathbb{Y}_{{\varepsilon}}\rightarrow \overline{\mathbb{R}}$ is a cost functional (CF _ε ),

(B ₂):

$\mathbb{Y}_{{\varepsilon}}$ is a space of states,

(B ₃):

$\mathbb{U}_{{\varepsilon}}$ is a space of controls,

(B ₄):

$\Xi_{{\varepsilon}}\subset\left\{(u_{{\varepsilon}},y_{{\varepsilon}})\in \mathbb{U}_{{\varepsilon}}\times \mathbb{Y}_{{\varepsilon}}\ :\ u\in U_{{\varepsilon}}, I_{{\varepsilon}}(u,y)<+\infty\right\}$ is a set of all admissible pairs linked by some state equation (SE _ε ).