Quadratic Lyapunov Functions

• Quadratic Lyapunov functions are defined as V(x)=xᵀPx with P symmetric positive definite, serving as certificates for stability by ensuring a decrease along trajectories.
• They are employed using LMIs and SDPs to verify stability across linear, switched, hybrid, and stochastic systems, ensuring tractable computational certification.
• These functions underpin convergence analysis in optimization, control synthesis, and region-of-attraction expansion, making them vital for complex dynamical systems.

A quadratic Lyapunov function is a real-valued function of the form $V(x) = x^\top P x$, where $x \in \mathbb{R}^n$ and $P = P^\top \succ 0$. Such functions are foundational in the analysis of stability, performance, and control for linear and nonlinear dynamical systems. They serve as certifying functions that decrease along system trajectories, thereby demonstrating stability, convergence, or other sought-after properties. Quadratic Lyapunov functions play a particularly central role in linear system theory, consensus dynamics, switched and hybrid systems, stochastic processes, and optimization algorithms. Their structure enables efficient verification via linear matrix inequalities (LMIs), admits geometric interpretation via ellipsoidal level-sets, and underpins many converse and synthesis results in control and applied mathematics.

1. Quadratic Lyapunov Functions: Definition and Canonical Settings

The classical quadratic Lyapunov function is defined for an equilibrium $x^* = 0$ as $V(x) = x^\top P x$, with $P \in \mathbb{R}^{n \times n}$ symmetric positive definite. For the ordinary differential equation (ODE) $\dot{x} = f(x)$ or the discrete-time map $x_{k+1} = \psi(x_k)$, $V$ is a Lyapunov function in a domain $D$ if:

• $V(x) > 0$ for $x \neq 0$, $V(0) = 0$
• $\dot{V}(x) < 0$ for all $x \in D \setminus \{0\}$ (continuous time)
• $V(\psi(x)) - V(x) < 0$ for all $x \in D$ (discrete time)

In the linear time-invariant (LTI) setting, $\dot{x} = A x$, the time derivative simplifies to $\dot{V}(x) = x^\top (P A + A^\top P) x$, so $P A + A^\top P \prec 0$ ensures global asymptotic stability. For discrete-time $x_{k+1} = A x_k$, the condition is $A^\top P A - P \prec 0$.

The solution space for $P$ is characterized by finite-dimensional LMIs, making the existence check, and in many cases synthesis, computationally tractable.

A critical extension pertains to systems where the state-space is partitioned or the vector field is nonsmooth—switched, hybrid, and piecewise-affine systems—prompting consideration of piecewise and max/min-composed quadratic Lyapunov functions (Dehghan et al., 2014, Adjé, 2015, Angeli et al., 2016, Pasquini et al., 2019).

2. Structural Results: Special Conic and Symmetry Properties

For systems preserving invariant cones, specialized quadratic Lyapunov functions arise with reduced parameterization and special geometric structure. For a continuous-time LTI system $ẋ = A x$ that preserves a proper, self-dual, homogeneous cone $C \subset \mathbb{R}^n$, if $A$ is Hurwitz, then there exists $P \in \mathrm{Aut}(C)$, symmetric and positive definite, such that $V(x) = x^\top P x$ is a global Lyapunov function with $P A + A^\top P \prec 0$. Notably, the complexity of specifying $P$ scales only linearly with $n$, as $P$ is determined (modulo scale) by the choice of a ray in $C$ (Dalin et al., 2023).

Two canonical cases:

• For $C = \mathbb{R}_+^n$, $\mathrm{Aut}_0(C)$ consists of diagonal positive matrices, so diagonal $P$ suffice.
• For the Lorentz (ice-cream) cone, $$K_n = \{x \in \mathbb{R}^n : \|x_{1:n-1}\|_2 \leq x_n, x_n \geq 0\}$$, $\mathrm{Aut}_0(K_n)$ consists of explicit matrix exponentials determined by a vector $b \in \mathbb{R}^{n-1}$. This yields explicit families of Lyapunov functions for Lorentz-invariant systems.

The approach is constructive: via the potential function $\varphi(x) = \int_{y \in C} e^{-x^\top y} dy$ and subsequent derivation of $P(x)$ as a Jacobian-based automorphism, followed by specialization to diagonal (orthant) and full (Lorentz) symmetry cases (Dalin et al., 2023).

3. Quadratic Lyapunov Functions in Algorithmic and Large-Scale Settings

Quadratic Lyapunov functions form the backbone of tight convergence analysis for first-order optimization algorithms, including gradient descent, heavy-ball, and accelerated methods. In the quadratic case, recent works have shown that standard and novel quadratic Lyapunov functions can be constructed via Schur decompositions or as solutions to small semidefinite programs (SDPs), yielding tight contraction rates (Taylor et al., 2018, Merkulov et al., 2023, Fercoq, 19 Nov 2024).

Key advances include:

• Reformulation of first-order methods as state-space recurrences where Schur-decomposition admits simple coordinate-wise quadratic Lyapunov functions profoundly linked to the system's spectral properties (Merkulov et al., 2023).
• Automated discovery of the optimal quadratic Lyapunov function guaranteeing the best possible linear rate, by posing and solving a performance-estimation saddle-point SDP, leveraging function interpolation and IQC conditions (Taylor et al., 2018, Fercoq, 19 Nov 2024).

In stochastic processing networks or consensus systems, local quadratic Lyapunov functions, often associated with individual subsystems or network nodes, can be aggregated via positive weights to yield global quadratic Lyapunov functions, enabling drift negativity and establishing positive Harris recurrence in Markov models (Dieker et al., 2012, Mangesius et al., 2014).

4. Piecewise, Path-Complete, and Hierarchical Quadratic Lyapunov Frameworks

For switching, piecewise-affine, and hybrid systems, single quadratic Lyapunov functions can be too restrictive. Piecewise-quadratic functions—maxima (or minima) of several quadratic forms—provide strictly more powerful certificates (Dehghan et al., 2014, Angeli et al., 2016, Adjé, 2015, Sel et al., 17 Jul 2025, Pasquini et al., 2019).

• Piecewise Quadratic Lyapunov Functions: Functions of the form $V(x) = \max_j x^\top P_j x$, with $P_j \succ 0$, tailored to local regions or modes. Feasibility is encoded by families of BMI or LMI constraints ensuring negativity of the derivative (or difference) in each partition and across switches (Dehghan et al., 2014, Adjé, 2015).
• Path-Complete Lyapunov Functions: Systems of quadratic Lyapunov inequalities structured by labeled graphs, where each node is a quadratic form and each edge encodes a Lyapunov decrease condition for a system mode. Such criteria guarantee the existence of non-smooth "common" Lyapunov functions constructed as min-max compositions of the pieces (Angeli et al., 2016).
• Hierarchies for LTV Systems: By lifting the original state-space to Kronecker or concatenated products, general polynomial Lyapunov functions can be constructed from quadratic forms in high-dimensional lifted spaces. This allows for the characterization of invariant sets and reachable value bounds for LTV systems via quadratic LMIs in the lifted coordinates (Abdelraouf et al., 23 Jan 2024).

In nonlinear and uncertain systems, partially quadratic or piecewise quadratic Lyapunov functions parameterized through SOS programming enable computationally tractable stability certification, particularly in systems where the state-space dimension is high but only lower-dimensional "center" variables exhibit slow dynamics (Jones et al., 2022, Sel et al., 17 Jul 2025).

5. Numerical and Algorithmic Construction: LMIs, SDP, and Policy Iteration

Verification and synthesis of quadratic Lyapunov functions are amenable to convex optimization. The existence of a quadratic Lyapunov function for $\dot{x} = A x$ requires the solution $P \succ 0$ to $P A + A^\top P \prec 0$, an LMI. For piecewise or path-complete constructions, families of LMIs for each region, mode, or edge must be satisfied (Dalin et al., 2023, Dehghan et al., 2014, Adjé, 2015, Angeli et al., 2016).

• Semidefinite and Policy Iteration: In PWA and hybrid systems the combination of LMIs with policy iteration over template parameters yields provably tight overapproximations of invariant or reachable sets (Adjé, 2015).
• SOS Programming: For partially quadratic or locally polynomial Lyapunov functions, sum-of-squares techniques enable the verification of positivity and negativity-of-derivative properties on regions of interest, reducing to semidefinite programming problems in the coefficients (Jones et al., 2022).
• Data-driven/Learning Approaches: For systems with a complicated feedback controller (e.g. neural networks), Lyapunov functions can be "learned" iteratively by alternating SDP candidate search with MILP-based counterexample generation, ensuring decrease constraints are satisfied everywhere (Chen et al., 2020).
• Validated numerics: Computer-assisted procedures can verify quadratic Lyapunov domains systematically using interval arithmetic, automated eigenvalue enclosure, and box-wise region partitioning, providing rigorous locality of Lyapunov certificates around equilibria (Matsue et al., 2016).

6. Quadratic Lyapunov Functions Beyond Finite Dimensionality

In infinite-dimensional settings—for example, linear control systems on Banach or Hilbert spaces—quadratic Lyapunov functions extend as functionals $V(x) = \langle P x, x \rangle$, with $P$ a bounded, positive, self-adjoint operator. Under suitable analyticity and dissipativity conditions (e.g., the semigroup is similar to a contraction), coercive quadratic ISS Lyapunov functions always exist and can be constructed explicitly or semi-explicitly in terms of operator semigroup integrals. However, the existence of ISS alone does not imply the existence of any coercive quadratic Lyapunov functional without these extra structural assumptions (Mironchenko et al., 2023).

For stochastic differential equations such as piecewise Ornstein-Uhlenbeck diffusions, common quadratic Lyapunov functions (CQLFs) for pairs of drift matrices provide a unifying mechanism for establishing positive recurrence and uniqueness of stationary distributions, with technical conditions linked to the spectral properties and M-matrix structure of the system (Dieker et al., 2011).

7. Extensions, Applications, and Computational Considerations

Quadratic Lyapunov functions underpin methodologies across control, optimization, and verification:

• Constructive control synthesis: In control design, quadratic Lyapunov functions yield tractable synthesis conditions for feedback laws, as in LQR, robust and $H_\infty$ control.
• Model reduction and positivity: When special structure (e.g., positive systems) is present, diagonal quadratic Lyapunov functions reduce computational complexity and admit interpretations as certificates for positivity invariance (Dalin et al., 2023).
• Nonstandard Finite Differences: For numerical integration of dissipative ODEs, discretization methods can be constructed to exactly preserve a quadratic Lyapunov function and maintain global asymptotic stability and positivity for any step size (Hoang, 2023).
• Adaptive and optimal trajectory generation: In nonlinear trajectory optimization, the use of full (non-diagonal) quadratic Lyapunov functions as Control-Lyapunov Functions (CLFs), parameterized via eigendecomposition, improves optimality over standard diagonal approaches (Nurre et al., 26 Aug 2024).
• Region of Attraction (ROA) expansion: Piecewise-quadratic Lyapunov methods, especially when combined with coordinate transformations, substantially enlarge certified ROA for nonlinear systems compared to single-ellipsoid (global quadratic) methods (Sel et al., 17 Jul 2025).

The interplay between analytic tractability, geometric insight, and computational feasibility ensures quadratic Lyapunov functions remain primary tools for the analysis, certification, and design of a wide range of complex dynamical systems.