Lyapunov-Based Stability Constraints

• Lyapunov-based stability constraints are mathematical conditions that guarantee exponential decay of a system's energy even with state and input limitations.
• They leverage convex SOS and SDP formulations to convert continuous control specifications into robust, verifiable stability conditions.
• Iterative algorithms enlarge the estimated region of attraction, ensuring rigorous certification of safety in nonlinear and hybrid control scenarios.

Lyapunov-based stability constraints formalize and certify the evolution of nonlinear, hybrid, or constrained control systems by encoding the decrease of a Lyapunov function in the presence of state and input constraints, quantified regions of attraction, disturbances, and structural system features. Modern developments have established rigorous, computationally viable recipes for certifying stability under realistic scenarios, including polyhedral input bounds, polynomial or neural Lyapunov candidates, set-invariance conditions, reach-avoid-stay specifications, and time-critical convergence. These constraints, often encoded as sum-of-squares (SOS) or semidefinite programs (SDPs), underpin controller synthesis, verification, and performance estimation in advanced safety-critical and robustness-driven applications.

1. Principles of Lyapunov-based Stability Constraints Under Constraints

A Lyapunov-based stability constraint requires the existence of a function $V(x)$ (the Lyapunov function) satisfying specific positivity and decay properties along the system trajectories, even in the presence of input or state constraints. For a control-affine system

$\dot{x} = f(x) + g(x)u, \quad u \in U,$

where $U$ is typically a convex polytope (input constraint), the constraints on $V(x)$ are:

• Positivity and coercivity: $V(x) - \epsilon\|x\|^{2\alpha} \in \Sigma[x]$, $V(0) = 0$, i.e., $V(x)$ is positive definite and radially unbounded.
• Existential decrease over constrained control: For all $x$ in a certified region $\Omega_\rho = \{x\,|\,V(x)<\rho\}$ (excluding the origin), there must exist a control $u \in U$ such that

$$L_f V(x) + L_g V(x) u \le -\kappa_V V(x), \quad \kappa_V > 0,$$

where $L_f V = \partial V/\partial x \cdot f(x)$ and $L_g V = \partial V/\partial x \cdot g(x)$.

Such constraints ensure the exponential decay of $V(x)$ along closed-loop trajectories within $\Omega_\rho$, giving both stabilization and an inner approximation of the region of attraction.

2. Convex SOS and SDP Encoding of Stability Constraints

When the input set $U = \operatorname{ConvHull}\{u^1, ..., u^m\}$ is a convex polytope, the decrease condition is affine in $u$ and can be certified over the vertices of $U$ without loss of generality: $$L_f V(x) + L_g V(x)u^i \le -\kappa_V V(x), \ \forall i = 1, ..., m.$$ The existential constraint $$\forall x \in \Omega_\rho \setminus \{0\}, \exists u \in U$$ is converted to finitely many robust constraints over $i=1,\ldots, m$.

The SOS S-procedure encodes the implication

$$\bigwedge_{i=1}^m (L_fV + L_g V u^i + \kappa_V V \ge 0) \implies (V(x) - \rho)(x^\top x) \ge 0$$

by searching for $\lambda_i(x), \lambda_0(x)$ SOS multipliers so that

$$(1+\lambda_0(x))(V(x) - \rho)(x^\top x) - \sum_{i=1}^m \lambda_i(x) [L_f V(x) + L_g V(x) u^i + \kappa_V V(x)] \in \Sigma[x]$$

and each $\lambda_i(x) \in \Sigma[x]$ (SOS).

The full SDP involves coefficients of $V(x)$, multipliers $\lambda_i(x)$, their Gram matrices, and scalar $\rho > 0$. Feasibility certifies a region $\Omega_\rho$ where the system is exponentially stabilizable under input constraints (Dai et al., 2022).

3. Region of Attraction Estimation and Algorithmic Enlargement

To enlarge the certified region, an iterative algorithm alternates between:

• Embedding an ellipsoid $$\mathcal{E}_d = \{x\,|\, (x - x_E)^\top S_E (x - x_E) \le d\}$$ in $\Omega_\rho$;
• For fixed $V, \rho$, maximizing $d$ such that the ellipsoid is contained in $\Omega_\rho$ via an SOS certificate;
• For fixed ellipsoid, updating multipliers $\lambda_i(x)$ via an SOS program;
• For fixed multipliers, updating $V, \rho$ to further enlarge the region of attraction.

Termination occurs when $d$ no longer increases appreciably. This algorithm produces an inner approximation to the stabilizable set, certified by the polynomial CLF and the input constraints (Dai et al., 2022).

4. Performance Guarantees and Theoretical Rigour

The feasibility of the SOS S-procedure directly certifies the nonnegativity of the Lyapunov decrease on the specified semi-algebraic set, yielding:

• Exponential convergence rate: $V(x)$ decays at rate $\kappa_V$ for all admissible trajectories in $\Omega_\rho$.
• Rigorous inner approximation: The certified region $\Omega_\rho$ is an inner approximation to the region of attraction.
• Polynomial data assumption: Guarantees are exact when $f(x), g(x), U$ are polynomial and SOS degrees are chosen sufficiently high (by Putinar’s Positivstellensatz).
• Nonconvexity handling: The only nonconvex terms (bilinear in $V, \lambda_i$) are treated via alternating SOS programs, converging in practice to locally optimal certificates (Dai et al., 2022).

5. Practical Implementation Steps

The entire synthesis framework can be summarized as follows:

Step	Constraint/Operation	Mathematical Form/SOS Encoding
1. Parametrize $V(x)$	Degree-$d$ polynomial with $V(0)=0$	$V(x) - \epsilon (x^\top x)^{\alpha} \in \Sigma[x]$
2. Handle input limits	$U=$ convex hull, reduce decrease constraint to vertex evaluations	$L_fV(x) + L_gV(x) u^i \le -\kappa_V V(x)$
3. Encode as SOS	S-procedure with SOS multipliers $\lambda_i(x), \lambda_0(x)$	See section above
4. Formulate SDP	Gram matrix variables for all SOS polynomials, scalar $\rho$	Convex feasibility (or maximize $\rho$)
5. Enlarge region	Iteratively optimize inscribed ellipsoid and update $V, \rho, \lambda_i$	Alternating convex programs

Solvers such as SOSTOOLS, YALMIP, or custom SOS-SDP code are typically used for numerical implementation. The approach does not require a nominal controller synthesis, streamlining the certification process and reducing conservatism compared to traditional methods that pair CLF and explicit controller design (Dai et al., 2022).

6. Extensions and Connections to Related Frameworks

• Barrier certificates and safety: The same SOS programming structure supports Control Barrier Function (CBF) synthesis, enabling joint stability and invariance specification.
• State and Input Generalizations: Polyhedral, quadratic, or semi-algebraic state/input sets can be incorporated directly as constraints in the SOS program.
• Alternate controller synthesis: The approach enables both hard-constrained (online QP/SDP) stabilizing feedback and explicit offline controller extraction from the multipliers.
• Integration with region growth: The iterative approach for $\Omega_\rho$ is compatible with methods for reaching outer approximations of maximal stabilizable regions (Dai et al., 2022).

7. Scope, Limitations, and Computational Aspects

This convex-SOS framework for Lyapunov stability constraints is exact for polynomial dynamics and inputs when the SOS degrees are sufficiently high, a guarantee rooted in algebraic geometry (Positivstellensatz). The main practical limitation is scalability—the complexity increases rapidly with system dimension and Lyapunov polynomial degree. Bilinearities between the Lyapunov polynomial and SOS multipliers are resolved via alternating minimization, reaching local but not necessarily global optima. Nonetheless, these methods enable rigorous, systematic certification and enlargement of regions of attraction under explicit actuator constraints, and they underpin a wide range of safety-critical and robust control designs (Dai et al., 2022).