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State Constrained Optimal Control Problems With Control On The Acceleration. Applications To Kinetic Mean Field Games

Yves Achdou

TL;DR

The paper addresses deterministic optimal control with acceleration as the control under state constraints, developing a variational framework to handle boundary singularities that impede PDE-based characterizations. It first obtains explicit 1D solutions for the special case $n=1$, $\\ell=0$, $g=0$ via an auxiliary variational problem, and then derives explicit forms and regularity for the full problem, including a detailed analysis of the value function and optimal trajectories. In higher dimensions with $q=2$, it adapts via boundary charts to prove closed-graph properties and singularity structure, providing a robust foundation for relaxed equilibria in kinetic mean field games with state constraints. The work culminates in existence results for relaxed mean field equilibria under general assumptions, highlighting the practical relevance for constrained dynamics in multi-agent systems. Overall, the paper delivers (i) explicit 1D formulas, (ii) a variational approach to boundary effects, (iii) closed-graph results in multiple dimensions, and (iv) existence of relaxed MFG equilibria with state constraints.

Abstract

Relying on the careful study of a related problem in the calculus of variations, we study a class of optimal control problems in which the control lies on the acceleration, with state constraints on the position variable. In dimension one, we find explicit formulas in the special case when the running cost is a power of the acceleration (in absolute value) and the terminal cost is zero. For more general costs or/and higher dimensions, we study the singularities of the value function. We also prove the closedness (in the C 1 topology) of the graph of the multivalued mapping which maps a point in the state space to the set of optimal trajectories which start from this point. A consequence of the latter is the existence, under general assumptions, of relaxed equilibria for a class of kinetic mean field games with state constraints.

State Constrained Optimal Control Problems With Control On The Acceleration. Applications To Kinetic Mean Field Games

TL;DR

The paper addresses deterministic optimal control with acceleration as the control under state constraints, developing a variational framework to handle boundary singularities that impede PDE-based characterizations. It first obtains explicit 1D solutions for the special case , , via an auxiliary variational problem, and then derives explicit forms and regularity for the full problem, including a detailed analysis of the value function and optimal trajectories. In higher dimensions with , it adapts via boundary charts to prove closed-graph properties and singularity structure, providing a robust foundation for relaxed equilibria in kinetic mean field games with state constraints. The work culminates in existence results for relaxed mean field equilibria under general assumptions, highlighting the practical relevance for constrained dynamics in multi-agent systems. Overall, the paper delivers (i) explicit 1D formulas, (ii) a variational approach to boundary effects, (iii) closed-graph results in multiple dimensions, and (iv) existence of relaxed MFG equilibria with state constraints.

Abstract

Relying on the careful study of a related problem in the calculus of variations, we study a class of optimal control problems in which the control lies on the acceleration, with state constraints on the position variable. In dimension one, we find explicit formulas in the special case when the running cost is a power of the acceleration (in absolute value) and the terminal cost is zero. For more general costs or/and higher dimensions, we study the singularities of the value function. We also prove the closedness (in the C 1 topology) of the graph of the multivalued mapping which maps a point in the state space to the set of optimal trajectories which start from this point. A consequence of the latter is the existence, under general assumptions, of relaxed equilibria for a class of kinetic mean field games with state constraints.

Paper Structure

This paper contains 24 sections, 16 theorems, 70 equations.

Table of Contents

  1. Introduction
  2. Description of the optimal control problem and known facts
  3. Notation
  4. Setting and assumptions
  5. Known facts
  6. Explicit formulas for the value function in \ref{['eq:6']} in the case $n=1$, $\ell=0$ and $g=0$
  7. An auxiliary variational problem
  8. Formulation of the auxiliary variational problem and characterization of its optimal value
  9. Variations of $I(\theta, w)$ with respect to $w$ for a fixed $\theta \in \left (\frac{|x|}{v}, \frac{(2q-1) |x|}{ (q-1) v} \right ]$
  10. Study of $\inf_{\theta \in (0,T] } I (\theta,w)$ for a fixed $w\in [0,v)$
  11. The optimal control problem
  12. The state constrained optimal control problem \ref{['eq:6']} in the case $n=1$
  13. Properties of the value function
  14. Closed graph properties
  15. The state constrained optimal control problem \ref{['eq:6']} in the case $n>1$ and $q=2$
  16. ...and 9 more sections

Key Result

Lemma 1

Consider a compact subset $\Theta$ of $\Xi^{\rm ad}$ (see eq:5) which has the following property: for all sequences $(x^i, v^i)_{i\in {\mathbb N}}$ with values $\Theta$ such that $\lim_{i\to +\infty} (x^i, v^i)=(x, v)\in \Theta$, the following holds: if $x\in \partial \Omega$ and $v\cdot n(x)=0$, t with the convention that if $x^i\in \partial \Omega$ and $v^i \cdot \nabla d (x^i) =v^i\cdot n(x^i)

Theorems & Definitions (41)

  • Remark 1.1
  • Lemma 1
  • Proposition 2
  • Remark 1.2
  • Proposition 1
  • Proof 1
  • Remark 2.1
  • Lemma 2
  • Proof 2
  • Lemma 3
  • ...and 31 more