Appendix D — Hybrid Dynamical Systems Background

Open source document →

1. #1

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

   $$\boxed{\frac{d{h}_i}{dt}=-\sum _{j:(i,j)\in E}{\nabla }_{h_i}{\mathcal{E}}_{F^{\circ }}(h)}$$
2. #2

   SPLIT guard condition:

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

   MERGE guard condition:

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

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

   $$x(t^{\ast })\in G_{q,q^{\prime }}\text{and}\dot{x}(t^{\ast })/\perp T_{x(t^{\ast })}{G}_{q,q^{\prime }}$$
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:

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