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Exponential Turnpike Theorems for Nonlinear Deterministic Meanfield Optimal Control Problems

Benoît Bonnet-Weill, Giovanni Colombo, Denis Shishmintsev, Emmanuel Trélat

Abstract

In this article, we establish exponential turnpike theorems for a class of nonlinear deterministic meanfield optimal control problems. We carry out our analysis simultaneously in the so-called Lagrangian and Eulerian frameworks. In the Lagrangian setting, the problem is lifted to a Hilbert space of random variables, and we prove an exponential turnpike theorem by combining first-order optimality conditions, a second-order expansion of the lifted Hamiltonian, and an operator Riccati diagonalization argument. In the Eulerian setting, we derive intrinsic KKT conditions for the static constrained problem, and show how the Eulerian second-order hypotheses split into a horizontal part, transferred by unitary conjugation to the lifted space, and a vertical part which reduces to uniform pointwise stabilizability and detectability conditions on multiplication operators. This yields an exponential turnpike theorem in the Wasserstein space for optimal Pontryagin triples. Along the way, we %provide explicit the link between Wasserstein Hessians and their Lagrangian lifts, and provide several remarks clarifying the role of occupation measures, local Eulerian minimizers, and control constraints in our results.

Exponential Turnpike Theorems for Nonlinear Deterministic Meanfield Optimal Control Problems

Abstract

In this article, we establish exponential turnpike theorems for a class of nonlinear deterministic meanfield optimal control problems. We carry out our analysis simultaneously in the so-called Lagrangian and Eulerian frameworks. In the Lagrangian setting, the problem is lifted to a Hilbert space of random variables, and we prove an exponential turnpike theorem by combining first-order optimality conditions, a second-order expansion of the lifted Hamiltonian, and an operator Riccati diagonalization argument. In the Eulerian setting, we derive intrinsic KKT conditions for the static constrained problem, and show how the Eulerian second-order hypotheses split into a horizontal part, transferred by unitary conjugation to the lifted space, and a vertical part which reduces to uniform pointwise stabilizability and detectability conditions on multiplication operators. This yields an exponential turnpike theorem in the Wasserstein space for optimal Pontryagin triples. Along the way, we %provide explicit the link between Wasserstein Hessians and their Lagrangian lifts, and provide several remarks clarifying the role of occupation measures, local Eulerian minimizers, and control constraints in our results.
Paper Structure (11 sections, 16 theorems, 166 equations)

This paper contains 11 sections, 16 theorems, 166 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Random variables and Lagrangian lifts.
  4. First-order optimality conditions for the static problems
  5. KKT conditions for the static Lagrangian problem
  6. KKT conditions for the static Eulerian problem
  7. Exponential turnpike theorems
  8. Turnpike in the Lagrangian framework
  9. Turnpike in the Eulerian framework
  10. Conclusion and perspectives
  11. Proof of Proposition \ref{['prop:WassHess']}

Key Result

Proposition 2.3

Suppose that $\Phi : \mathcal{P}_2(\mathbb{R}^d) \to \mathbb{R}$ is Fréchet differentiable in the sense of Definition def:WassGrad. Then, its lift $\widetilde{\Phi} : L^2_{\mathbb{P}}(\Omega,\mathbb{R}^d) \to \mathbb{R}$ is Fréchet differentiable in the usual sense, and it holds that for every $X \in L^2_{\mathbb{P}}(\Omega,\mathbb{R}^d)$.

Theorems & Definitions (41)

  • Definition 2.1: Fréchet differentiable functions over $\mathcal{P}_2(\mathbb{R}^d)$
  • Remark 2.2: On the definition of Wasserstein derivative
  • Proposition 2.3: First-order differentiability and gradient formula
  • proof
  • Definition 2.4: Twice continuously differentiable functions over $\mathcal{P}_2(\mathbb{R}^d)$
  • Proposition 2.5: Second-order differentiability and Hessian formula
  • proof
  • Lemma 2.6: Composition operators and the representation of lifted Hessian
  • proof
  • Remark 2.7: Concerning Hypotheses \ref{['hyp:H']}
  • ...and 31 more