Nonsmooth-Aware Stability Frameworks

• Nonsmooth-Aware Stability Frameworks are rigorous methods that certify existence, uniqueness, and invariance of solutions in systems with discontinuous or piecewise-smooth dynamics.
• They employ projection rules and Filippov inclusions to handle state variables at physical boundaries, ensuring trajectories remain within defined domains.
• The framework is broadly applicable across climate, biological, and engineering models where state constraints lead to nonsmooth phenomena, ensuring global stability.

A nonsmooth-aware stability framework is a rigorous analytic structure designed to certify the existence, uniqueness, and long-term invariance of solutions to dynamical systems whose governing equations or constraints possess nonsmooth features, such as kinks, corners, or boundary-induced discontinuities. These frameworks extend classical smooth dynamical systems analysis to cases where either the state dynamics or physical constraints induce upper-semicontinuous, piecewise-smooth, or set-valued vector fields. Central to such approaches is the principled handling of physical boundaries—such as climate states being constrained between fully glaciated and ice-free configurations—and the use of generalized solution concepts and stability mechanisms that account for the inherent nonsmoothness of the system. The formulation in the context of climate energy-balance and carbon-cycle models provides a prototypical and systematic account of this methodology (Barry et al., 2014).

1. Physical and Mathematical Foundation

The archetype for a nonsmooth-aware stability framework is provided by energy-balance climate models with hard physical boundaries:

• The principal variables are an "ice-line latitude" $\eta(t)\in[0,1]$ that evolves due to feedback mechanisms, and an atmospheric carbon proxy $A(t)\in\mathbb{R}$ whose evolution is modulated by slow geological and chemical processes.
• In the interior ($0<\eta<1$), the governing equations comprise a slow CO$_2$ drift

$\dot{A} = \delta\,(\eta-\eta_c)$

and a fast ice-line law $\dot{\eta} = h(A,\eta)$, where $h$ is a physically smooth function encoding albedo and solar flux feedbacks.

• Physical boundaries at $\eta=0$ (fully glaciated) and $\eta=1$ (ice-free) dictate that unconstrained ODE evolution may attempt to exit the physical domain, which requires specialized boundary motion rules.

2. Boundary Motion and Nonsmooth Projection Rules

A defining feature of this framework is the explicit role assigned to boundaries:

• Whenever the dynamics drive $\eta$ toward or beyond its permissible domain $[0,1]$, the evolution law is "projected" to prevent unphysical states:

$$f(A,\eta) = \begin{cases} 0 & \eta = 0 \text{ and } h(A,0) < 0 \ 0 & \eta = 1 \text{ and } h(A,1) > 0 \ h(A,\eta) & \text{otherwise} \end{cases}$$

• This "projection rule" imposes a hybrid nonsmooth right-hand side: at the boundaries, $\dot{\eta}$ is forced to zero if the ODE vector field points outward, effectively "sticking" the solution on the boundary until the restoring drift in $A$ brings $h(A,\eta)$ back to a value that allows $\eta$ to re-enter the interior.
• Alternatively, an equivalent differential inclusion formulation, in the sense of Filippov, continuously assigns a set-valued vector field outside $[0,1]$ that always points inward—ensuring equivalence between the two nonsmooth solution concepts.

3. Existence, Uniqueness, and Forward Invariance

The main stability results rest on exploiting the structure created by these projection rules:

• Existence and uniqueness of solutions are established by combining classical ODE results in the interior (where the vector fields are $C^1$) with careful tracking of crossing times when trajectories hit or leave the boundaries.
• The resulting solutions may contain at most finitely many "corner points"—moments when the state sticks to or releases from the boundary, marking transitions between different vector fields.
• A crucial property is the forward invariance of the physical region: no solution can ever escape the physically meaningful strip $\mathbb{R}\times [0,1]$, even under arbitrary initial conditions within this domain. This means every point on a trajectory, for all forward times, remains within the set allowed by physical constraints.

4. Nonsmooth Stability as a Model-Invariant Property

The invariance induced by the nonsmooth framework serves as a robust stability property:

• Boundaries act as impenetrable gates: when $\eta$ reaches $0$ (glaciated) or $1$ (ice-free), the feedback dynamics responsible for rapid evolution ("ice-albedo feedback") are deactivated, and only the slow $A$-dynamics remain.
• On the boundary, the system slides along a segment until the slow variable $A$ evolves to a critical value, at which point $h$ changes sign, releasing the trajectory back into the interior. This mechanism directly mirrors natural phenomena, such as a snowball climate eventually deglaciating as CO$_2$ accumulates.
• In phase-space, the set of slow quasi-steady equilibria—determined by the (degenerate) critical manifold $\{g(A,\eta)=0, h(A,\eta)=0\}$—attracts the evolution, ensuring (in a singular perturbation sense) that the long-term dynamics adhere to a lower-dimensional, normally hyperbolic slow manifold.

5. Monotonicity and Comparison in the Absence of Smooth Lyapunov Functions

Although no traditional Lyapunov function is constructed, stability is established via comparison and monotonicity:

• The $A$ variable acts formally as a Lyapunov-like coordinate along the physical boundaries: on $\eta=0$, $\dot{A} = \delta(0-\eta_c)<0$ ensures $A$ decreases strictly, guaranteeing finite-time exit from the boundary; a similar argument holds for $\eta=1$.
• These monotonic flows prevent indefinite "sticking" at any boundary, providing a key argument for both uniqueness and long-term adherence to the physical strip.

6. Structural Blueprint for General Piecewise-Smooth Systems

The framework distilled here is presented as a general template for nonsmooth-aware stability in any system where state variables are physically or mathematically constrained:

• Decomposition of the dynamics into smooth vector fields on open domains separated by switching or boundary manifolds.
• Clear, physically or mathematically motivated rules (projection, Filippov sliding, or other formulations) for what occurs when the state contacts a boundary or switching surface.
• Regularity (local Lipschitz or $C^1$ on each side, upper-semicontinuity across interfaces) ensuring the well-posedness of the hybrid ODE or inclusion.
• Monotonicity or sign-conditions on boundary flows that force forward invariance and guarantee release from sliding regimes within finite time.
• Identification of attracting critical manifolds (slow sets or quasi-equilibria) that govern and stabilize the long-term dynamics, independent of the presence of nonsmooth events.

7. Applicability and Scope Across Nonsmooth Systems

The nonsmooth-aware stability framework outlined is broadly applicable to a range of models beyond the extended Budyko climate case:

• Any climate, biological, engineering, or physical model in which a state variable is strictly bounded by domain constraints (e.g., volume fractions, concentrations, physical positions) may be analyzed within this paradigm.
• The framework guarantees that physically nonsensical behavior—such as a system escaping from its domain—cannot occur, and that the nonsmooth (e.g., corner, kink, stick-slip) events are tractable, finite, and incorporated in the existence-uniqueness theory.
• This approach, fundamentally geometric and structural rather than dependent on smooth Lyapunov functions, provides a robust analytic toolkit for studying the global-in-time behavior, boundary-induced quasi-equilibria, and long-term stability of nonsmooth dynamical systems (Barry et al., 2014).

In sum, nonsmooth-aware stability frameworks provide a rigorous, physically-motivated analytic architecture for systems constrained by boundaries or governed by piecewise-smooth vector fields, ensuring global well-posedness, invariance, and stabilization even in the absence of classical smoothness throughout phase-space.