A small-gain theorem for fixed-time stability

Fixed time: one deadline for every starting point

A finite-time stable system reaches its equilibrium in finite time, but that time can grow without bound as the starting point moves away. A fixed-time stable system has a single deadline that works for every starting point.

The standard test is a Lyapunov function that decreases with two powers at once: a power below one that finishes the job near the origin, and a power above one that pulls far-away states in quickly. If \[ \dot V \le -c_1 V^{p_1} - c_2 V^{p_2}, \]\[ p_1 \in (0,1),\ \ p_2 > 1, \] then the origin is fixed-time stable and \[ T(x_0) \le \frac{1}{c_1(1-p_1)} + \frac{1}{c_2(p_2-1)} \quad \text{for every } x_0. \]

Try it on the scalar system \(\dot x = -\sgnp{x}{p} - \sgnp{x}{q}\), where \(\sgnp{x}{a} = |x|^a \operatorname{sgn}(x)\). Without the second term it is only finite-time stable.

Two fixed-time systems in a loop

The papers study the interconnection

Each subsystem treats the other's state as an input, alongside its own external input \(u_i\). The only requirement on the parts is that each one is fixed-time input-to-state stable on its own:

Here \(\K^P_\infty\) is the class of sums of powers \(\sum_i c_i s^{p_i}\), and \(\K^{-P}_\infty\) the class of their inverses. Fixed-time input-to-state stability means \(|x(t)| \le \beta(|x_0|,t) + \varrho(|u|_\infty)\) with \(\beta \in \KL_{\mathrm{FxT}}\): after a fixed time, the state is bounded by a gain of the input alone. With \(u \equiv 0\) this is fixed-time stability, which is the result of the ACC 2026 paper.

The proof builds one Lyapunov function for the whole loop, \[ V(x) = \max\{V_{11}(x_1),\ V_{22}(x_2),\ V_{21}(x_2),\ V_{12}(x_1)\}, \] where each \(V_{ij}\) is a subsystem's Lyapunov function rescaled through the gains and the power \(\sigma_\lambda(s) = s^\lambda\). For \(\lambda\) large enough every rescaled piece is still a fixed-time ISS Lyapunov function, so whichever piece is active in the maximum decreases at a fixed-time rate.

Why the inverse gains matter. If both gains are sums of powers with different exponents, \(\hat\gamma_1 \circ \hat\gamma_2(s) < s\) must fail either as \(s \to 0\) or as \(s \to \infty\). Allowing one gain to be the inverse of a power sum turns the condition into \(\hat\gamma_1(s) < \hat\gamma_2^{-1}(s)\): a comparison of two power sums, which can hold for every \(s\). The next example shows this comparison at work.

How much coupling the loop can take

Theorem 2 of the ACC paper applies Theorem 1 to two fixed-time systems with polynomial coupling,

\[ \dot x = f(x) + \varepsilon_1\big(\sgnp{y}{p} + \sgnp{y}{q}\big), \]\[ \dot y = g(y) - \varepsilon_2 \sgnp{x}{r}, \]

with \(f(x) = -x^{1/3} - x^3\) and \(g(y) = -\sgnp{y}{1/2} - \sgnp{y}{2}\). With \(V = x^2\) and \(W = y^2\), the subsystems satisfy \(2xf(x) = -2|x|^{4/3} - 2|x|^{4}\) and \(2yg(y) = -2|y|^{3/2} - 2|y|^{3}\), so \(m_1 = m_2 = 1\), \(\alpha_2 = 3/4\) and \(\beta_2 = 3/2\). The theorem certifies fixed-time stability when

\[ pr,\ qr \in [\alpha_2, \beta_2] \]\[ \text{and} \quad 0 < \varepsilon_1^r \varepsilon_2 < \frac{m_1^r m_2}{2^r L_r^{1/2}}, \]\[ L_r = \max(1, 2^{r-1}). \]

The exponent window is what replaces the usual requirement \(pr = qr = 1\): exponents only need to sit near one. Below, the left plot compares the two gains from the proof, \(\gamma_1(s)\) and \(\gamma_2^{-1}(s)\); the small-gain condition asks the blue curve to stay under the amber one for every \(s\). The exponents \(p = 2\) and \(q = 2.5\) are fixed as in the paper.

With inputs, the state settles into a band set by the input

Example (19) of the journal paper adds external inputs to a system with several coupling terms:

\[ \begin{aligned} \dot x &= -x^{1/3} - x^3 + 0.1\big(\sgnp{y}{2} + \sgnp{y}{2.5}\big) + u_1(t), \\ \dot y &= -\sgnp{y}{1/2} - \sgnp{y}{9/4} - 0.3\sgnp{x}{4/9} + 0.2\sgnp{x}{5/8} + u_2(t), \end{aligned} \]

with \(u_1(t) = \sin(1.7t) + \tfrac12\sin(5.1t)\) and \(u_2(t) = \tfrac45\cos(2.3t) + 0.35\sin(6.7t)\). It meets the exponent conditions of Proposition 1 in the paper: every product of a coupling exponent from one equation with one from the other, \(\eta_{1,i}\eta_{2,j} \in \{0.89, 1.11, 1.25, 1.56\}\), lies in \([p_2, q_2] = [3/4, 13/8]\), so the proposition guarantees fixed-time ISS once the couplings are small enough. Scale the inputs to see the guarantee: whatever the starting point, after a fixed time the state stays in a band whose size depends only on the input.

Feedback optimization without separating time scales

A plant \(\dot x = f(x,u)\) settles at \(x = h(u)\) for each constant input. The goal is to steer it to the minimizer of a cost \(\phi_\theta(x,u)\) that drifts over time, \(\dot\theta = \varepsilon_0 \Pi(\theta)\). Substituting the steady state gives the target \(\varphi(\theta) = \arg\min_u \phi_\theta(h(u), u)\).

The usual design makes the controller much slower than the plant, so the plant can be treated as already settled. Here the controller runs at full speed and reads the measured state \(x\) in place of \(h(u)\):

\[ \dot u = -\mathcal F_\xi\big(\mathcal G_\theta(x,u)\big), \]\[ \mathcal F_\xi(s) = \frac{s}{|s|^{\xi_1}} + \frac{s}{|s|^{\xi_2}}, \]\[ \xi_1 \in (0,1),\ \xi_2 < 0, \]

where \(\mathcal G_\theta(x,u) = H(u)^\top \nabla \phi_\theta(x,u)\) is the gradient of the cost along the steady-state map.

Here \(\mu\) is the strong convexity of the reduced cost, \(K\) bounds how much the plant state perturbs the gradient, and \(\ell\) and \(\gamma\) describe the plant. In the paper's example (section IV-C), \(\ell = 1.1\), \(\gamma = 7/5\), \(K = 1.339\) and \(\mu = 3.37 > K(\ell + \gamma) = 3.35\). The slider sets \(\varepsilon_0\); the paper shows 0, 0.3 and 2.

Nash seeking with a plant in the loop

Now several players share one plant. Player \(i\) chooses \(u_i\) and pays \(J_i(x,u)\), which depends on the plant state. Each player updates using only its own partial derivative, again with no time-scale separation from the plant:

\[ \dot u_i = -\mathcal F_\xi\big(\mathcal G_i(x,u)\big), \]\[ \mathcal G_i(x,u) = \begin{bmatrix} \tfrac{\partial h(u)}{\partial u_i}^{\!\top} & e_i^\top \end{bmatrix} \nabla J_i(x,u). \]

The bound grows with \(\sqrt N\): with more players, the curvature of the game has to outweigh the plant's influence by more. In the paper's two-player example (section V-B), \(\mu = 2.605 > \sqrt2\, K(\ell+\gamma) = 1.80\). The shading below is the potential function the players jointly descend, and the star is the Nash equilibrium.

The papers

@article{tang2027smallgain,
  title   = {A Lyapunov-Based Small-Gain Theorem for Fixed-Time {ISS}:
             Theory, Optimization, and Games},
  author  = {Tang, Michael and Krstic, Miroslav and Poveda, Jorge I.},
  journal = {IEEE Transactions on Automatic Control},
  year    = {2027},
  doi     = {10.1109/TAC.2026.3738248}
}

@inproceedings{tang2026smallgain,
  title     = {A Lyapunov-Based Small-Gain Theorem for
               Fixed-Time Stability},
  author    = {Tang, Michael and Krstic, Miroslav and Poveda, Jorge I.},
  booktitle = {American Control Conference (ACC)},
  year      = {2026},
  url       = {https://ieeexplore.ieee.org/document/11615748}
}