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Pattern-preserving optimal control problems with increasing time horizon

Matteo Della Rossa, Lorenzo Freddi

TL;DR

This work develops a Γ-convergence framework to guarantee that optimal-control patterns observed on finite horizons persist on the infinite horizon, even in the presence of state constraints. By decomposing the cost and establishing Γ-convergence of the joint functionals $F_T=J_T+\chi_{\mathcal{A}}$ to a limiting $F_\infty$, the authors prove pattern-preservation results under coercivity w.r.t. the state $x$ or under compact state constraints. They address the analytical challenges of unbounded time domains by careful choice of state and control topologies and by introducing a tails-replacement argument for recovery sequences. The framework is illustrated with applications to switched systems and epidemic SIR models, highlighting Bang-Bang control structures that carry over to the infinite horizon and offering a rigorous basis for long-term control strategies.

Abstract

We establish a general framework that guarantees the preservation of optimal control patterns as the time horizon $[0,T]$ increases and becomes unbounded. A concept of pattern-preserving family of optimal control problems is introduced and the goal is achieved by analyzing the $Γ$-convergence of the corresponding variational formulations as $T\to\infty$. Special attention is given to scenarios involving state constraints. To illustrate the results, examples and applications are provided, with particular focus on switched systems and epidemic control.

Pattern-preserving optimal control problems with increasing time horizon

TL;DR

This work develops a Γ-convergence framework to guarantee that optimal-control patterns observed on finite horizons persist on the infinite horizon, even in the presence of state constraints. By decomposing the cost and establishing Γ-convergence of the joint functionals to a limiting , the authors prove pattern-preservation results under coercivity w.r.t. the state or under compact state constraints. They address the analytical challenges of unbounded time domains by careful choice of state and control topologies and by introducing a tails-replacement argument for recovery sequences. The framework is illustrated with applications to switched systems and epidemic SIR models, highlighting Bang-Bang control structures that carry over to the infinite horizon and offering a rigorous basis for long-term control strategies.

Abstract

We establish a general framework that guarantees the preservation of optimal control patterns as the time horizon increases and becomes unbounded. A concept of pattern-preserving family of optimal control problems is introduced and the goal is achieved by analyzing the -convergence of the corresponding variational formulations as . Special attention is given to scenarios involving state constraints. To illustrate the results, examples and applications are provided, with particular focus on switched systems and epidemic control.

Paper Structure

This paper contains 14 sections, 19 theorems, 121 equations.

Table of Contents

  1. Introduction
  2. $\Gamma$-convergence and pattern-preserving sequences
  3. Running assumptions and main results
  4. State equations: basic Carathéodory-Lipschitz assumptions
  5. State equations: global existence and convergence of solutions
  6. Intrinsically coercive control systems
  7. Uniformly bounded systems
  8. Linear control systems
  9. Optimal control problems with increasing time horizons
  10. Applications and examples
  11. Optimal switching control problems
  12. Optimal control of SIR models
  13. Conclusions
  14. Local Sobolev function spaces

Key Result

Theorem 2.4

Consider a sequence $\mathcal{F}_k:\mathcal{U}\times \mathcal{X}\to \overline \mathbb{R}$ that $\Gamma^-_{\rm seq}(\mathcal{U}\times\mathcal{X})$-converges to $\mathcal{F}$ and let $(u_k,x_k)$ be a minimizing sequence The following propositions hold.

Theorems & Definitions (57)

  • Remark 2.1
  • Definition 2.2
  • Definition 2.3
  • Theorem 2.4: variational property
  • proof
  • Remark 2.5
  • Definition 2.6: pattern-preserving family
  • Theorem 2.7: pattern preservation
  • proof
  • Remark 2.8: $p=\infty$, limit of piecewise constant controls
  • ...and 47 more