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Collective steering in finite time: controllability on $\text{GL}^+(n,\mathbb{R})$

Mahmoud Abdelgalil, Tryphon T. Georgiou

TL;DR

This work studies collective steering of $N$ identical linear agents via a common feedback to realize a rearrangement of their configuration, formulated as controllability on the identity component of $GL^+(n,\mathbb{R})$. By recasting the problem as a right-invariant bilinear system $\dot{\Phi}_t=(A+B K_t)\,\Phi_t$ with $\det(\Phi_t)>0$, the authors establish controllability on $GL^+(n,\mathbb{R})$ under the Kalman rank condition and, further, strong controllability by constructing motion primitives and leveraging an SPD-factorization (Ballantine's result) to decompose targets into at most five factors. A topological obstruction shows that a universal, continuous feedback gain depending on problem data cannot exist for this class of systems, though well-behaved continuous laws exist for related linear and Lyapunov dynamics. The paper also extends the discussion to orientation-preserving diffeomorphisms, linking controllability insights to optimal transport and holonomy concepts, and provides an illustrative numerical example to demonstrate the constructive path-building approach.

Abstract

We consider the problem of steering a collection of n particles that obey identical n-dimensional linear dynamics via a common state feedback law towards a rearrangement of their positions, cast as a controllability problem for a dynamical system evolving on the space of matrices with positive determinant. We show that such a task is always feasible and, moreover, that it can be achieved arbitrarily fast. We also show that an optimal feedback control policy to achieve a similar feat, may not exist. Furthermore, we show that there is no universal formula for a linear feedback control law to achieve a rearrangement, optimal or not, that is everywhere continuous with respect to the specifications. We conclude with partial results on the broader question of controllability of dynamics on orientation-preserving diffeomorphisms.

Collective steering in finite time: controllability on $\text{GL}^+(n,\mathbb{R})$

TL;DR

This work studies collective steering of identical linear agents via a common feedback to realize a rearrangement of their configuration, formulated as controllability on the identity component of . By recasting the problem as a right-invariant bilinear system with , the authors establish controllability on under the Kalman rank condition and, further, strong controllability by constructing motion primitives and leveraging an SPD-factorization (Ballantine's result) to decompose targets into at most five factors. A topological obstruction shows that a universal, continuous feedback gain depending on problem data cannot exist for this class of systems, though well-behaved continuous laws exist for related linear and Lyapunov dynamics. The paper also extends the discussion to orientation-preserving diffeomorphisms, linking controllability insights to optimal transport and holonomy concepts, and provides an illustrative numerical example to demonstrate the constructive path-building approach.

Abstract

We consider the problem of steering a collection of n particles that obey identical n-dimensional linear dynamics via a common state feedback law towards a rearrangement of their positions, cast as a controllability problem for a dynamical system evolving on the space of matrices with positive determinant. We show that such a task is always feasible and, moreover, that it can be achieved arbitrarily fast. We also show that an optimal feedback control policy to achieve a similar feat, may not exist. Furthermore, we show that there is no universal formula for a linear feedback control law to achieve a rearrangement, optimal or not, that is everywhere continuous with respect to the specifications. We conclude with partial results on the broader question of controllability of dynamics on orientation-preserving diffeomorphisms.

Paper Structure

This paper contains 12 sections, 12 theorems, 88 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Collective Controllability
  3. Optimal control in feedback form?
  4. Brockett's approach
  5. Controllability on $\text{GL}^+(n,\mathbb{R})$
  6. Strong Collective Controllability
  7. A topological obstruction to the continuity of a universal feedback gain
  8. Continuity of control laws to specifications
  9. Representative cases
  10. Illustrative example
  11. On the controllability of orientation-preserving diffeomorphisms
  12. Concluding Remarks

Key Result

Proposition 1

Assume that the pair $(A,B)$ satisfies the Kalman rank condition, and that for a given $t_{\mathrm{fn}}>0$ a matrix $K_c$ has been selected so that eq:pole-placement-periodic holds. If $\Phi{_{\mathrm{fn}}}\in\text{GL}^+(n,\mathbb{R})$ is specified so that then $\Phi_t^\star$ belongs to $\text{GL}^+(n,\mathbb{R})$ for all $t\in[0,t_{\mathrm{fn}}]$.

Figures (2)

  • Figure 1: The first segment, i.e. $\Phi_{t}^{1,\star}$ with $t\in[0,t_s]$, in the piecewise smooth curve \ref{['eq:piecewise-smooth-trajectory-exmp']}, which starts from $I$ and ends at $\Phi_1$.
  • Figure 2: The second segment, i.e. $\Phi_{t-t_s}^{1,\star}$ with $t\in[t_s,2t_s]$, in the piecewise smooth curve \ref{['eq:piecewise-smooth-trajectory-exmp']}, which starts from $\Phi_1$ and ends at $\Phi_2$.

Theorems & Definitions (29)

  • Example 1
  • Proposition 1
  • proof
  • Remark 1
  • Theorem 1
  • proof
  • Remark 2
  • Proposition 2
  • proof
  • Remark 3
  • ...and 19 more