Appendix D — Hybrid Dynamical Systems Background

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1. #1

   Within a fixed topology , the continuous dynamics are driven by Yang--Mills gradient flow:

   $$\boxed{\frac{dh_i}{dt} = -\sum_{j: (i,j)\in E} \nabla_{h_i} \mathcal{E}_{F^\circ}(h)}$$
2. #2

   SPLIT guard condition:

   $$G_{\mathrm{SPLIT}} := \{(x,t) : \exists F \in \mathfrak{F}_t \text{ with } \phi_t(F) \text{ crosses } \theta \text{ from below}\}$$
3. #3

   MERGE guard condition:

   $$G_{\mathrm{MERGE}} := \{(x,t) : \exists F_1, F_2 \in \mathfrak{F}_t \text{ with } \phi(F_1 \cup F_2) \text{ crosses } \theta \text{ from above}\}$$
4. #4

   Definition D.1 (Transversal crossing): The trajectory is transversal to guard at time if:

   $$x(t^*) \in G_{q,q'} \quad \text{and} \quad \dot{x}(t^*) \not\perp T_{x(t^*)} G_{q,q'}$$
5. #5

   Define the total Lyapunov function for the hybrid system:

   $$V(q, x) := \mathcal{E}(x) + \lambda \cdot (\text{topological cost})$$
6. #6

   Theorem D.1 (Hybrid stability): The hybrid evolution satisfies:

   $$\frac{dV}{dt}\bigg|_{\text{continuous}} \le -\alpha \|\dot{x}\|^2 \quad (\alpha > 0)$$