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Impulsive Control on Invariant Surfaces

C. C. Silva, J. Marao, A. Figueiredo, T. M. Rocha Filho

Abstract

An impulsive feedback-adaptive control is developed in order to drive trajectories of a dynamical system towards an invariant manifold with fixed and spaced impulsive controls. The approach requires the explicit knowledge of the set of equations defining the invariant manifold and is based on the concept of stability exponents of invariant manifolds.

Impulsive Control on Invariant Surfaces

Abstract

An impulsive feedback-adaptive control is developed in order to drive trajectories of a dynamical system towards an invariant manifold with fixed and spaced impulsive controls. The approach requires the explicit knowledge of the set of equations defining the invariant manifold and is based on the concept of stability exponents of invariant manifolds.
Paper Structure (9 sections, 74 equations, 4 figures)

This paper contains 9 sections, 74 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Basic Notations and Definitions
  3. Semi-Invariants and Stability Exponents
  4. Impulsive Systems and Convergence Control
  5. Applications
  6. The Lorenz system: Convergence towards a Fixed Point
  7. Synchronization of two Lorenz systems
  8. Parallel Impulsive Control
  9. Concluding Remarks

Figures (4)

  • Figure 1: Left panel: time evolution $||{\bf I}(t)||$ for the Lorenz system with no control (without impulses) and two different impulsive controls with $\kappa=3.0$ and $\kappa=7.0$. Right panel: cumulative number of impulses as a function of time $t$ for $\kappa=3.0$ and $\kappa=7.0$.
  • Figure 2: Stability exponent $D_S$ as a function of the coupling parameter $c$.
  • Figure 3: Left panel: norm of $||{\bf I}||$ as a function of time for the cases without control and for impulses with $\alpha=0.1$ and $\alpha=0.4$. Right panel: $\log ||{\bf I}||$ for the cases with an impulsive control.
  • Figure 4: Left panel: evolution of the cumulative number of cases as a proportion of the total population. Right panel: epidemic curves (new cases per day) as a proportion to the total population.