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Behavioral Inequalities

Soutrik Bandyopadhyay, Debasattam Pal, Shubhendu Bhasin

TL;DR

The paper proposes behavioral inequalities to model dynamical systems defined by temporal inequalities of the form $H(\sigma,\sigma^{-1}) w \le g$, enabling safety-aware and bound-constrained descriptions. It establishes a necessary and sufficient feasibility condition via a generalized Farkas' lemma on the adjoint of the polynomial shift operator and provides a slack-variable parametrization to characterize all feasible trajectories. The framework handles mixed equalities and inequalities and yields constructive solution representations through slack trajectories and unimodular transformations. Two practical examples—safety-aware dynamical systems and dynamic inventory management—demonstrate feasibility testing, solution parametrization, and potential for constraint-aware data-driven control in safety-critical applications.

Abstract

We introduce behavioral inequalities as a way to model dynamical systems defined by inequalities among their variables of interest. We claim that such a formulation enables the representation of safety-aware dynamical systems, systems with bounds on disturbances, practical design limits and operational boundaries, etc. We develop a necessary and sufficient condition for the existence of solutions to such behavioral inequalities and provide a parametrization of solutions when they exist. Finally, we show the efficacy of the proposed method in two practical examples.

Behavioral Inequalities

TL;DR

The paper proposes behavioral inequalities to model dynamical systems defined by temporal inequalities of the form , enabling safety-aware and bound-constrained descriptions. It establishes a necessary and sufficient feasibility condition via a generalized Farkas' lemma on the adjoint of the polynomial shift operator and provides a slack-variable parametrization to characterize all feasible trajectories. The framework handles mixed equalities and inequalities and yields constructive solution representations through slack trajectories and unimodular transformations. Two practical examples—safety-aware dynamical systems and dynamic inventory management—demonstrate feasibility testing, solution parametrization, and potential for constraint-aware data-driven control in safety-critical applications.

Abstract

We introduce behavioral inequalities as a way to model dynamical systems defined by inequalities among their variables of interest. We claim that such a formulation enables the representation of safety-aware dynamical systems, systems with bounds on disturbances, practical design limits and operational boundaries, etc. We develop a necessary and sufficient condition for the existence of solutions to such behavioral inequalities and provide a parametrization of solutions when they exist. Finally, we show the efficacy of the proposed method in two practical examples.

Paper Structure

This paper contains 12 sections, 4 theorems, 49 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Behavioral Systems Theory
  4. Infinite Dimensional Vector Spaces
  5. Behavioral Inequalities
  6. Feasibility of Behavioral Inequalities (Existence of Solutions)
  7. Mixed Behavioral Inequalities and Equations
  8. Characterization of Solutions of the Behavioral Inequalities
  9. Practical Examples
  10. Safety-aware Dynamical Systems
  11. Dynamic Inventory Management Problem
  12. Concluding Remarks

Key Result

Lemma 1

Consider two real valued vector spaces $\mathcal{X}$, $\mathcal{Y}$ with an inner product $\left\langle \cdot, \cdot \right\rangle$ defined on $\mathcal{Y}$. Let $A: \mathcal{X} \to \mathcal{Y}$ be a linear operator and $b \in \mathcal{Y}$ be a constant vector. Let the vector space $\mathcal{Y}$ be

Figures (1)

  • Figure 1: Infeasibility of behavioral inequality constraints discussed in Example \ref{['examp:infeasible_lti']}. The blue region shows the state constraints. The quiver plot for the dynamics of the system is shown in gray. We observe that trajectories originating in the constraint region eventually violate the safety constraint.

Theorems & Definitions (14)

  • Definition 1: Behavioral Inequalities
  • Example 1
  • Lemma 1: Generalized Farkas' Lemma clark2006necessary
  • proof
  • Lemma 2: Adjoint of polynomial shift operator
  • proof
  • Theorem 1: Feasibility condition for behavioral inequality
  • proof
  • Example 2
  • Theorem 2
  • ...and 4 more