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HJ inequalities involving Lie brackets and feedback stabilizability with cost regulation

Giovanni Fusco, Monica Motta, Franco Rampazzo

Abstract

With reference to an optimal control problem where the state has to approach asymptotically a closed target while paying a non-negative integral cost, we propose a generalization of the classical dissipative relation that defines a Control Lyapunov Function to a weaker differential inequality. The latter involves both the cost and the iterated Lie brackets of the vector fields in the dynamics up to a certain degree k greater than or equal to 1, and we call any of its (suitably defined) solutions a degree-k Minimum Restraint Function. We prove that the existence of a degree-k Minimum Restraint Function allows us to build a Lie-bracket-based feedback which sample stabilizes the system to the target while regulating (i.e., uniformly bounding) the cost.

HJ inequalities involving Lie brackets and feedback stabilizability with cost regulation

Abstract

With reference to an optimal control problem where the state has to approach asymptotically a closed target while paying a non-negative integral cost, we propose a generalization of the classical dissipative relation that defines a Control Lyapunov Function to a weaker differential inequality. The latter involves both the cost and the iterated Lie brackets of the vector fields in the dynamics up to a certain degree k greater than or equal to 1, and we call any of its (suitably defined) solutions a degree-k Minimum Restraint Function. We prove that the existence of a degree-k Minimum Restraint Function allows us to build a Lie-bracket-based feedback which sample stabilizes the system to the target while regulating (i.e., uniformly bounding) the cost.
Paper Structure (17 sections, 6 theorems, 121 equations)

This paper contains 17 sections, 6 theorems, 121 equations.

Table of Contents

  1. Introduction
  2. Notation and preliminaries
  3. Degree-$k$ sample stabilizability with regulated cost
  4. Admissible trajectories and global asymptotic controllability with regulated cost
  5. Degree-k feedback generator
  6. Sampling processes
  7. Degree-$k$ sample stabilizability with regulated cost
  8. The main result
  9. Degree-$k$ Hamilton-Jacobi dissipative inequality
  10. Main result
  11. Proof of Theorem \ref{['Th_strong']}
  12. A degree-$k$ feedback generator and some preliminary estimates
  13. Estimating $U$ increments when $\mathcal{V}(x)$ has degree $\ell\le k$
  14. Stabilizing $\mathfrak{d}$-scaled $\mathcal{V}$-sampling processes
  15. The cost boundedness property
  16. ...and 2 more sections

Key Result

Lemma 2.6

Assume (H1)- (H2) and fix $\tilde{R}>0$. Then, there exist $\bar{\delta}>0$ and $\omega>0$ such that for any $x \in \overline{\mathcal{B}(\mathcal{T},\tilde{R})\setminus\mathcal{T}}$, any control label $(B, \mathbf{g},\text{\rm sgn})\in \mathcal{F}^{(k)}$ of degree $\ell$ and switch number $\mathfr defined on the whole interval $[0,t]$ and satisfyingWe use the notation '$\text{\rm sgn} \, B( \mat

Theorems & Definitions (37)

  • Definition 1.1
  • Definition 2.1: Admissible controls, trajectories, and costs
  • Definition 2.2: Global asymptotic controllability with regulated cost
  • Definition 2.3: Control label
  • Definition 2.4: Oriented control
  • Example 2.5
  • Lemma 2.6
  • Remark 1
  • Definition 2.7: Degree-$k$ feedback generator
  • Definition 2.8: Multiflow
  • ...and 27 more