关于分段有限时间Lyapunov函数（piecewise finite-time Lyapunov function）的微分包含（Differential inclusion）应用及全局有限时间稳定性（finite-time stability）证明的技术问询

Hey there, great question—this is a nuanced spot where standard asymptotic stability results don’t map directly, so let’s break this down step by step to adapt differential inclusions to your finite-time scenario.

First, let’s recap the core of your problem: you’ve got piecewise Lyapunov functions where each segment already guarantees finite-time convergence (with that square-root decay rate), rather than asymptotic stability. You want to use differential inclusions to prove the entire system is finite-time stable, even though most existing work starts from asymptotic segments and works up to finite-time behavior.

Key Adjustments for Finite-Time Segments #

The main shift from asymptotic to finite-time lies in the Lyapunov decay condition we need to enforce via differential inclusions. For asymptotic stability, each segment satisfies $\dot{V}_i \leq -k_i V_i$; for finite-time, your segments already satisfy $\dot{V}_i \leq -k_i V_i^{1/2}$ (with $k_i > 0$) in their respective regions $\Omega_i$. Here’s how to wrap this into a differential inclusion framework:

1. Ensure Continuity of the Piecewise Lyapunov Function
   This is non-negotiable. Your global Lyapunov function $V(x) = V_i(x)$ for $x \in \Omega_i$ must be continuous across region boundaries. If $V$ jumps at $\partial\Omega_i \cap \partial\Omega_j$, you’ll lose control of the right derivative (critical for differential inclusions) and can’t guarantee consistent decay. So first, verify $V_i(x) = V_j(x)$ for all $x$ on shared boundaries.

2. Enforce a Uniform Decay Rate Across All Segments
   For global finite-time stability, you can’t have $k_i$ dropping to zero in any segment. You need a universal constant $k > 0$ such that every $k_i \geq k$. This ensures the square-root decay rate is consistent everywhere, which is required to bound the convergence time globally (finite-time stability demands a fixed upper bound on convergence time for all initial states).

3. Frame the Differential Inclusion for Finite-Time Decay
   For your piecewise smooth system, the state dynamics can be written as a differential inclusion: $\dot{x} \in \mathcal{F}(x)$, where $\mathcal{F}(x)$ contains the vector field $f_i(x)$ for whichever region $\Omega_i$ $x$ belongs to (and the convex combination of adjacent fields on boundaries, if needed).

   For your Lyapunov function, we work with the right Dini derivative $D^+V(x)$ (since $V$ might not be differentiable everywhere). For all $x \neq 0$, you need to show:

   D^+V(x) = \limsup_{h \to 0^+} \frac{V(x + h f(x)) - V(x)}{h} \leq -k V(x)^{1/2}
   

   In each region $\Omega_i$, this holds because you already have $\dot{V}_i = \nabla V_i \cdot f_i \leq -k_i V_i^{1/2} \leq -k V^{1/2}$. On boundaries, since $V$ is continuous, the limsup will be bounded by the minimum decay rate of adjacent segments—still $\leq -k V^{1/2}$ thanks to your uniform $k$.

4. Apply the Finite-Time Lyapunov Theorem for Differential Inclusions
   There’s a direct extension of finite-time stability theory to differential inclusions: if you have a positive-definite, radially unbounded function $V$, and a constant $k > 0$, $0 < \alpha < 1$ (here $\alpha = 1/2$) such that $D^+V(x) \leq -k V(x)^\alpha$ for all $x \neq 0$, then the system is globally finite-time stable.

   Since you’ve already verified each segment meets the decay condition, and you’ve ensured continuity and uniform decay, this theorem applies directly to your global $V$ via the differential inclusion framework.

Why This Works (Even When Existing Studies Go the Other Way) #

Most existing work starts with asymptotic segments because it’s a weaker condition—proving finite-time behavior from asymptotic is harder. Your scenario is actually more straightforward: each segment already has the stronger finite-time decay, so you just need to "glue" them together properly (via continuity and uniform rates) to extend that behavior globally using differential inclusions.

The differential inclusion here acts as a way to formalize the piecewise dynamics without getting bogged down in switching rules or boundary behavior—by framing the derivative as a set of possible values, you can ensure the decay condition holds no matter which region the state is in (or transitioning between).

备注：内容来源于stack exchange，提问作者Yaosheng Deng