Convex Stability & Safety Conditions

• Convex stability and safety conditions are mathematically verifiable constraints that certify controlled system stability using convex formulations like LMIs, SDPs, and SOS.
• They integrate tools such as Lyapunov functions, barrier certificates, and chance constraints to handle uncertainties, nonlinearity, and stochastic dynamics.
• These methods are applied in areas such as autonomous driving, learning-based controllers, and robust positive systems, highlighting both practical implementations and inherent approximation challenges.

Convex stability and safety conditions are a set of rigorous, mathematically verifiable constraints that guarantee the stability of controlled dynamical systems and the satisfaction of safety or feasibility specifications, using convex—typically semidefinite or polyhedral—programming formulations. These conditions are crucial in modern control, planning, stochastic optimization, and learning-based feedback design as they ensure that the true, closed-loop system trajectories remain within prescribed safe sets (invariant regions, chance constraints, or risk envelopes), and that candidate controllers or solutions are robust to uncertainty, model mismatch, and disturbances. Convexity is essential since it enables tractable computation, reliable certification, and systematic scalability to high-dimensional or infinite-dimensional settings.

1. Theoretical Foundations of Convex Stability and Safety

The core methodology underlying convex stability and safety is the reduction of generally intractable controller synthesis or verification problems to convex programs—principally linear matrix inequalities (LMIs), semidefinite programs (SDPs), or second-order cone programs (SOCPs). This is realized by leveraging Lyapunov theory, barrier certificates, quadratic (or integral quadratic) constraints, sector bounds, and occupation measure formalism.

For continuous or discrete-time systems, the existence of a Lyapunov function whose decrease can be certified by a convex constraint (typically an LMI) immediately yields stability. For safety, invariant sets or constraints such as control barrier functions (CBFs) or probabilistic (chance) constraints are imposed, again forced into convex representations by restricting function classes (e.g., to sum-of-squares polynomials, quadratic forms, or convex set-indicator functions).

Convexity guarantees that local optimality is global, feasibility is efficiently checkable, and, in some cases, the entire safe region or region of attraction can be characterized as the feasible set of a single convex program (Yin et al., 2020, Dai et al., 2022, Colombino et al., 2015).

2. Lyapunov and Barrier Certificates via Convex Optimization

The principal tool for certifying stability is the Lyapunov function—a scalar function $V(x)$ that decreases along trajectories, typically enforced by imposing $V(x)$ positive definite and $dV/dt < 0$ (continuous time) or $V(x_{k+1}) < V(x_k)$ (discrete time). In systems with input or state constraints and nonlinearities (e.g., neural network controllers), sector- or slope-bounds for the activation functions are encoded as local quadratic constraints, which are then enforced using LMIs (Yin et al., 2020, Biazetto et al., 15 Nov 2025).

For example, in LTI plants with NN controllers, the Lyapunov decrease and the sector-bounds on the NN’s nonlinearities yield jointly convex LMIs in the Lyapunov parameter $P$ and auxiliary quadratic constraint multipliers (Yin et al., 2020). The volume of the certified region of attraction ($\{x : V(x) \leq 1\}$) can be directly maximized (via $\log\operatorname{det}$ criteria) inside the convex optimization (Yin et al., 2020, Biazetto et al., 15 Nov 2025).

Control barrier functions (CBFs) guarantee forward invariance of safety sets. Their synthesis can also be formulated via convex (SOS, SDP) constraints in polynomial systems, with the safety constraint (e.g., $\dot h(x,u) \geq -\alpha(h(x))$ for some class-$\mathcal{K}$ function $\alpha$) enforced pointwise or over polynomial input bounds (Dai et al., 2022).

Convexity ensures that the existence and numerical construction of these certificates is scalable and certifiable, yielding strong guarantees of closed-loop stability and safety.

3. Convex Feasibility and Safety in Stochastic and Uncertain Systems

Stochastic dynamics, measurement noise, and model uncertainty require extension to probabilistic or robust certificates. Chance constraints and risk contour formulations pose the problem: enforce that a safety violation happens with probability not exceeding a prescribed threshold.

Convex analysis of chance-constrained feasible sets determines explicit critical values ($\alpha^*$) above which the set is convex for general noise structures (elliptical, copula-based), establishing a “stability threshold” for convex safety-preserving design (Laguel et al., 2021). These results leverage transform-concavity of the safety margin and concavity of the noise law to ensure the superlevel set $S(\alpha)$ is convex. For instance, in linear-Gaussian models, the classic result is that the halfspace is convex if $\alpha\ge 1/2$ (Laguel et al., 2021).

Advanced frameworks use occupation measures and barrier certificates: the worst-case unsafe probability is formulated as an infinite-dimensional convex linear program in occupation measures, with the dual variable representing a continuous barrier certificate function (Miller et al., 1 Jan 2024). Sum-of-squares relaxations yield tractable, convergent SOS hierarchies for certification in polynomial (semi-algebraic) systems. This approach generalizes naturally to risk contours and arbitrary initial distributions.

GP-based and model-uncertain scenarios formulate stability and safety as second-order cone constraints (SOCPs), with tight probabilistic (high-probability) bounds on the effect of uncertainties imposed using the posterior mean and variance of GP-represented model errors (Castañeda et al., 2021, Hall et al., 2023).

4. Structural and Robust Convex Stability: LTI and Markov Systems

Convex stability and safety methods extend to structured, interconnected, or positive systems. For linear positive and positively dominated systems, necessary and sufficient conditions for robust stability under block-diagonal uncertainties reduce to testing a single diagonal-scaling LMI on the static gain, exploiting the fact that the structured singular value is exactly represented by its convex upper bound (Colombino et al., 2015). This yields robust Lyapunov functions and explicit safety margins, applied for example to wireless power control algorithms.

For finite-state controlled Markov chains, safe recurrent sets are computed as the support of the optimizer of a convex entropy-maximization program under linear balance (stationarity) and forbidden-state constraints (Arvelo et al., 2012). The resulting maximal safe recurrent set and a control law inducing invariant distributions supported exactly on this set are both certified via convex programming.

For dissipative systems or Lur’e-type interconnections, convex parametrizations of integral quadratic constraints (IQCs), including O’Shea-Zames-Falb multipliers with terminal cost, yield finite-horizon state and output bounds that are tractably optimized via LMI (Scherer, 2022).

5. Design Frameworks and Algorithmic Integration

Convex stability and safety conditions are systematically embedded in controller synthesis, learning, and online or adaptive methods:

• Imitation learning with guarantees: Neural network controllers are trained with stability/safety LMIs as hard constraints, using ADMM to alternate between policy learning and certificate search (Yin et al., 2020, Biazetto et al., 15 Nov 2025).
• Convex MPC and safety filters: Differential flatness or feedback-linearized models, with GP-uncertainty, are combined with convex QP (for trajectory planning) and SOCP (for safety filtering and high-probability CLF/CBF constraints) layers (Hall et al., 2023).
• Online robust control: Safety is ensured under adversarial disturbances by convex constraint tightening (“buffer zones”) derived from tail bounds on closed-loop Lyapunov contraction, enforced at each OGD (Online Gradient Descent) step as polyhedral feasibility conditions (Jiang et al., 29 Jan 2025).
• Sum-of-squares region maximization: Inner approximations of the stabilizable or safe sets are iteratively grown using SOS programs that alternate between inscribing ellipsoids and recomputing certificates (Dai et al., 2022).
• Motion planning: In convex spatio-temporal corridors, Bézier trajectory coefficients are placed at time samples of boundary functions, and a sufficient convex-hull condition guarantees full containment with gap decaying as $O(1/n^2)$ (Zhang et al., 2021).

The following table summarizes representative convex stability and safety frameworks:

Domain/Problem	Convex Condition	Program Type
LTI + NN control	Lyap/QC LMI	SDP/ADMM
Polynomial systems	SOS CLF/CBF	SDP (SOS)
Stochastic safety	Occup. measure barrier	Infinite-D / SOS SDP
Markov chain design	Stationary entropy max	Convex program
Robust positive LTI	Diag-scaled LMI	Small SDP
Online robust ctrl	Buffer zone polyhedra	Linear/SOCP at each step
Flat MPC + GP	CLF/CBF SOCP	QP+SOCP
IQC/dissipativity	Dynamic IQC LMI	LMI, terminal cost

6. Practical Implications and Limitations

Convex stability and safety conditions provide strong guarantees on the certified regions, with reliable computational tractability. The primary limitation remains the conservatism introduced by the chosen relaxations or sectors (particularly in S-procedure and sum-of-squares approaches), and by the need to approximate infinite-dimensional conditions (e.g., for PDEs, occupation measures, or high-order polynomials). For certain high-dimensional or hard nonlinear problems, scalability may require sparse or decomposable convexification strategies.

Further, the safety and feasibility guarantees are always limited to the fidelity of the underlying models and the quality of uncertainty characterizations (e.g., GP posterior confidence, sector bounds, or truncation order in PDEs). In some applications, feasibility is only certified on restricted sublevel sets or for truncated state representations, and convergence to true safe/stabilizable sets as the approximation order grows is subject to regularity and compactness assumptions (Biazetto et al., 15 Nov 2025, Dai et al., 2022, Miller et al., 1 Jan 2024).

7. Illustrative Applications Across Domains

Demonstrations in the literature show the breadth of convex stability and safety methods:

• Autonomous driving motion planning in time-varying, general-convex corridors, yielding smoother trajectories and smaller gaps than with classical corridors (Zhang et al., 2021).
• Learning-based controllers for both finite- and infinite-dimensional plants, including boundary stabilization of reaction-diffusion PDEs (Biazetto et al., 15 Nov 2025).
• Robust positive systems such as the Foschini–Miljanic power-control algorithm for wireless communication (Colombino et al., 2015).
• Online linear control under bounded adversarial disturbances, with sublinear cumulative regret (Jiang et al., 29 Jan 2025).
• Synthesis/verification of polynomial CLFs/CBFs under hard actuator limits, including iterative region maximization (Dai et al., 2022).
• Certified stochastic safety for SDEs, discrete-time Markov processes, and PDEs, using occupation-measure SOS techniques (Miller et al., 1 Jan 2024).

Practical implementations routinely leverage off-the-shelf SDP/SOCP solvers (e.g., MOSEK, CVX, YALMIP, ECOS) and symbolic or numerical tools for sum-of-squares programming and region-of-attraction computation.

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In summary, convex stability and safety conditions serve as a mathematically principled, computationally tractable foundation for certifying and synthesizing stable, safe control and optimization policies across a range of deterministic, stochastic, and learning-enabled systems, with a direct impact on scalability, robustness, and reliability in both theory and practice (Yin et al., 2020, Dai et al., 2022, Colombino et al., 2015, Laguel et al., 2021, Miller et al., 1 Jan 2024, Biazetto et al., 15 Nov 2025, Zhang et al., 2021, Jiang et al., 29 Jan 2025, Hall et al., 2023, Scherer, 2022, Arvelo et al., 2012, Castañeda et al., 2021, Reis et al., 2020).