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Pattern preservation in finite to infinite-horizon optimal control problems for dissipative systems

Matteo Della Rossa, Thiago Alves Lima, Lorenzo Freddi

Abstract

This paper focuses on infinite-horizon optimal control problems for dissipative systems and the relations to their finite-horizon formulations. We show that, for a large class of problems, dissipativity of the state equation, when a coercive storage function exists, implies that infinite-horizon optimal controls can be obtained as limits of the corresponding finite-horizon ones. This property is referred to as pattern preservation, or pattern-preserving property. Our analysis establishes a formal link between dissipativity theory and the variational convergence framework in optimal control, thus providing a concrete and numerically tractable condition for verifying pattern preservation. Numerical examples illustrate the effectiveness and limitations of the proposed sufficient conditions.

Pattern preservation in finite to infinite-horizon optimal control problems for dissipative systems

Abstract

This paper focuses on infinite-horizon optimal control problems for dissipative systems and the relations to their finite-horizon formulations. We show that, for a large class of problems, dissipativity of the state equation, when a coercive storage function exists, implies that infinite-horizon optimal controls can be obtained as limits of the corresponding finite-horizon ones. This property is referred to as pattern preservation, or pattern-preserving property. Our analysis establishes a formal link between dissipativity theory and the variational convergence framework in optimal control, thus providing a concrete and numerically tractable condition for verifying pattern preservation. Numerical examples illustrate the effectiveness and limitations of the proposed sufficient conditions.
Paper Structure (12 sections, 7 theorems, 108 equations, 1 figure)

This paper contains 12 sections, 7 theorems, 108 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries: pattern-preserving optimal control problems
  3. Admissible sets defined by controlled ODEs
  4. Sequences of control problems with increasing horizon
  5. A suitable re-formulation
  6. Dissipativity and pattern-preserving property
  7. Dissipativity and equi-coercivity
  8. Dissipativity and Gamma-Convergence
  9. Examples and Applications
  10. (Non-)Linear Quadratic Regulator
  11. Conclusions
  12. $\Gamma$-convergence

Key Result

Theorem 2.5

Let $p\in (1,\infty]$ and $\mathcal{U}=L^p((0,\infty),U)$ endowed with its weak$^\star$ topology. Consider a parametrized family of optimal control problems $\mathcal{F}_T:\mathcal{U}\times \mathcal{X}\to (-\infty,+\infty]$, $T\in (0,\infty]$, and suppose that there exists a sequence $0\leq T_k\to \ Then $\mathcal{F}_T$ is $(T_k)$-pattern preserving.

Figures (1)

  • Figure 1: Graphical representation of the singular-arc Pontryagin extremal $u_{s,T}$ in Example \ref{['ex:NONDISSIpative']} for $T=10$. The BOCOP toolbox can be used to numerically observe that these extremals are optimal, at least for large $T$, and this can be formally confirmed by explicit computation of the cost. The infinite-horizon control $u_{s, \infty}$ is obtained as limit, when $T\to \infty$, of such piecewise-constant functions, and its formal definition is in \ref{['eq:Infinite-HorizonControl']}.

Theorems & Definitions (25)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3: Pattern-preserving family
  • Example 2.4: Non patter-preserving sequence of OCPs
  • Theorem 2.5: pattern preservation
  • Remark 2.6: The choice of the state space and its topology
  • Lemma 2.7
  • Remark 2.8: Motivations behind the new formulation
  • Definition 3.1: Dissipativity
  • Remark 3.2: Brief history of dissipativity and its applications in optimal control
  • ...and 15 more