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Symplectic Geometry in Hybrid and Impulsive Optimal Control

William Clark, Maria Oprea

TL;DR

This work develops a geometric, symplectic framework for hybrid and impulsive optimal control by extending Pontryagin's maximum principle to Hamiltonian hybrid systems (HPMP) via an augmented differential. It proves that Hamiltonian jumps are symplectomorphisms and that the hybrid flow preserves the symplectic form and volume, enabling a global description of optimal trajectories as intersections of hybrid Lagrangian submanifolds. The paper introduces caustics and hybrid conjugate points, analyzes irregular executions such as beating and Zeno, and demonstrates these ideas with a bouncing-ball impact example and a synchronization problem for spiking neurons. The results provide a principled way to characterize optimal discontinuous trajectories and lay groundwork for numerical methods to handle irregular arcs, with potential applications in biology and robotics.†

Abstract

Hybrid dynamical systems are systems which undergo both continuous and discrete transitions. The Bolza problem from optimal control theory was applied to these systems and a hybrid version of Pontryagin's maximum principle was presented. This hybrid maximum principle was presented to emphasize its geometric nature which made its study amenable to the tools of geometric mechanics and symplectic geometry. One explicit benefit of this geometric approach was that the symplectic structure (and hence the induced volume) was preserved. This allowed for a hybrid analog of caustics and conjugate points. Additionally, an introductory analysis of singular solutions (beating and Zeno) was discussed geometrically. This work concluded on a biological example where beating can occur.

Symplectic Geometry in Hybrid and Impulsive Optimal Control

TL;DR

This work develops a geometric, symplectic framework for hybrid and impulsive optimal control by extending Pontryagin's maximum principle to Hamiltonian hybrid systems (HPMP) via an augmented differential. It proves that Hamiltonian jumps are symplectomorphisms and that the hybrid flow preserves the symplectic form and volume, enabling a global description of optimal trajectories as intersections of hybrid Lagrangian submanifolds. The paper introduces caustics and hybrid conjugate points, analyzes irregular executions such as beating and Zeno, and demonstrates these ideas with a bouncing-ball impact example and a synchronization problem for spiking neurons. The results provide a principled way to characterize optimal discontinuous trajectories and lay groundwork for numerical methods to handle irregular arcs, with potential applications in biology and robotics.†

Abstract

Hybrid dynamical systems are systems which undergo both continuous and discrete transitions. The Bolza problem from optimal control theory was applied to these systems and a hybrid version of Pontryagin's maximum principle was presented. This hybrid maximum principle was presented to emphasize its geometric nature which made its study amenable to the tools of geometric mechanics and symplectic geometry. One explicit benefit of this geometric approach was that the symplectic structure (and hence the induced volume) was preserved. This allowed for a hybrid analog of caustics and conjugate points. Additionally, an introductory analysis of singular solutions (beating and Zeno) was discussed geometrically. This work concluded on a biological example where beating can occur.

Paper Structure

This paper contains 16 sections, 12 theorems, 88 equations, 12 figures, 1 table, 1 algorithm.

Table of Contents

  1. Introduction
  2. Hybrid and impact dynamical systems
  3. Hybrid systems
  4. Impact systems
  5. Hybrid control systems
  6. Necessary conditions for optimality
  7. Hybrid Lagrangian submanifolds
  8. Lagrangian intersections
  9. Caustics
  10. Irregular executions
  11. Beating executions
  12. Zeno executions
  13. Application: synchronization for spiking neurons
  14. Conclusions
  15. Hybrid conjugate points
  16. ...and 1 more sections

Key Result

Theorem 1

A differential form $\alpha\in\Omega^*(M)$ is preserved along the hybrid flow if, and only if, $\mathcal{L}_X\alpha=0$ and where $\iota:{S}\hookrightarrow M$ is the inclusion and $i_X\alpha = \alpha(X,\cdot)$ is interior multiplication. Let $\mathcal{A}_\mathcal{H}\subset\Omega^*(M)$ be the set of all invariant differential forms. Then $\mathcal{A}_\mathcal{H}$ forms a $\wedge$-subalgebra that is

Figures (12)

  • Figure 1: The shortest path connecting points $A$ and $B$ while also touching the line $\sigma$.
  • Figure 2: When the $\mathrm{codim}(S)\geq 2$, resets become rare. In particular, generic trajectories will "miss" the guard. In this example, the guard is a single point in the plane and only a single arc resets - all others miss.
  • Figure 3: A hybrid system with the quasi-smooth dependence property. Although the flow fails to be smooth along the guard, the restriction to the open neighborhood $U$ is smooth when both $U$ and $\varphi^\mathcal{H}(t,U)$ are disjoint from $\mathcal{S}$.
  • Figure 4: Induced guards from Example \ref{['ex:resets']}. Left: A plot of the projected vector field $(\pi_Q)_*X_{H_1}$. As this vector field is nowhere transverse to the gaurd, the induced guard is empty, $S_{H_1}=\emptyset$. Right: A fiber of the induced guard, $S_{(x,y)}\subset T^*_{(x,y)}Q$, corresponding to $H_2$. The red hatched region corresponds to all outward pointing vectors rooted at the point $(x,y)\in\Sigma$.
  • Figure 5: An example of hybrid Lagrangian $\mathcal{L}\subset T^*M$clarkoprea.
  • ...and 7 more figures

Theorems & Definitions (43)

  • Definition 1: Hybrid Dynamical Systems
  • Remark 1
  • Definition 2: Hybrid Flow
  • Definition 3: Regular Hybrid Arc
  • Definition 4: Quasi-Smooth Dependence Property
  • Theorem 1: clark2023invariant
  • Definition 5: Augmented Differential
  • Definition 6
  • Remark 2
  • Example 1
  • ...and 33 more