Non-Standard Lyapunov Functional Analysis

• Non-standard Lyapunov functionals are augmented energy measures that incorporate tailored cross-terms and spatial weightings to achieve uniform exponential stability in systems with delays and degeneracies.
• They utilize multiplier techniques and weighted Sobolev spaces to manage boundary effects, non-uniform damping, and complex feedback mechanisms in distributed parameter systems.
• This approach facilitates explicit decay estimates and rigorous stability proofs, extending classical energy methods to a broader class of PDEs with singular or non-local features.

A non-standard Lyapunov functional is an augmented or tailored energy-like quantity constructed to facilitate the analysis of asymptotic stability, exponential decay, or robustness properties in evolution equations where the classical energy functional fails to yield definitive or sharp results. This concept is crucial in distributed parameter systems with geometric inhomogeneity, singular or degenerate coefficients, boundary delays, non-local dissipation, thermal effects, or complex feedback mechanisms. Non-standard functionals typically incorporate cross-terms, spatial weightings, or boundary integrals that exploit specific structural properties of the PDE and its control architecture.

1. Mathematical Motivation and Definition

A Lyapunov functional $V(t)$ is any non-negative, differentiable map on the solution trajectory of a PDE or ODE that strictly decreases along non-trivial orbits. For standard parabolic or hyperbolic systems, the total mechanical or thermomechanical energy is often sufficient. However, in systems with delays, degeneracies, non-local couplings, or non-self-adjoint operators, direct computation of the time derivative typically yields only non-uniform or insufficient decay.

A non-standard Lyapunov functional supplements the classical energy $E(t)$ by introducing problem-adapted cross-terms:

$L(t) = E(t) + \varepsilon G(t)$

where $G(t)$ may involve weighted spatial integrals, boundary quantities, transport variables, or multipliers. The technical requirements are:

• $C_0$-equivalence: $c_1 E(t) \leq L(t) \leq c_2 E(t)$ for some $c_1, c_2 > 0$,
• Dissipativity: $dL/dt \leq -\delta E(t)$ modulo manageable remainder terms,
• Suitably constructed $G(t)$ so that boundary or delay effects are absorbed or controlled.

This non-standard approach is strongly associated with the use of multiplier techniques and weighted Sobolev spaces, especially in problems of degenerate, non-uniform, or delayed damping (Siriki et al., 20 Dec 2025, V. et al., 2015).

2. Construction in Degenerate and Non-Uniform Systems

A prototypical application appears in boundary stabilization of degenerate Euler-Bernoulli beams under axial force and time delay. The governing equation incorporates non-uniform flexural rigidity $\sigma(x)$ with $\sigma(0)=0$ and axial force $q(x)$:

$$u_{tt}(x,t) + \partial_{xx}\big[\sigma(x)u_{xx}(x,t)\big] - \partial_x\big[q(x)u_x(x,t)\big] = 0.$$

Standard energy estimates fail due to degeneracy at $x=0$ and delayed boundary feedback.

The non-standard Lyapunov functional is:

$L(t) = E(t) + \varepsilon G(t),$

where

$$E(t) = \frac{1}{2}\left[ \int_0^1 \left( u_t^2 + \sigma u_{xx}^2 + q u_x^2 \right) dx + \gamma \tau \int_0^1 w^2 ds + \kappa_b u(1)^2 + \kappa_r u_x(1)^2 \right],$$

and

$$G(t) = \int_0^1 u_t(x,t)\left[2x\,u_x(x,t) + \frac{\Upsilon_{\sigma,q}}{2}\,u(x,t)\right] dx + \gamma \tau \int_0^1 e^{-2\tau s}u_t^2(1,t-\tau s)ds,$$

with $$\Upsilon_{\sigma,q} = \max \left\{ K_{\sigma}, q_2/q_0 \right\}$$, and $w(s,t)=u_t(1,t-s\tau)$ is an auxiliary transport variable encoding the boundary delay (Siriki et al., 20 Dec 2025).

The cross-term $G(t)$ is chosen so that its time derivative supplies the necessary control over degenerate spatial regions and delayed boundary terms, enabling absorption of trace estimates back into the bulk energy $E(t)$.

3. Multiplier Techniques and Cross-Terms

Multiplier methods systematically generate non-standard Lyapunov functionals. For example, in inhomogeneous Euler-Bernoulli beams with thermal effects (V. et al., 2015), the classical mechanical energy

$$E(t) = \frac{1}{2} \left\{ \int_0^L p(x)u_{xx}^2 dx + \int_0^L u_t^2 dx + \int_0^L \theta^2 dx \right\}$$

is insufficient to guarantee uniform exponential decay under inhomogeneous damping $q(x)$ and thermal coupling $\kappa \neq 0$. The augmented functional is

$$G(t) = E(t) + \varepsilon F_1(t), \quad F_1(t) = \int_0^L u u_t dx + \int_0^L q(x)u^2 dx,$$

with estimates $|F_1(t)| \leq \mu E(t)$ and $dF_1/dt = -2E(t) + R_1(t)$, where $R_1(t)$ is expressed in terms of controllable quadratic forms. The cross-terms in $F_1(t)$ exploit the damping profile and spatial structure, yielding full exponential stability (V. et al., 2015).

4. Weighted Sobolev Spaces and Functional Equivalence

Non-standard Lyapunov frameworks often require the ambient Hilbert space (state space) itself to be weighted or adapted to the degeneracy of the operator. For degenerate beam equations,

• $$V^2_{\sigma}(0,1)=\{ u \in H^1(0,1): \sqrt{\sigma} u_{xx} \in L^2 \}$$,
• $H^2_{\sigma}(0,1)=\{ u \in V^2_{\sigma}: u(0)=0 \}$.

Energy and Lyapunov constructions explicitly use norms inherited from such spaces. Functional equivalence lemmas guarantee the augmented $L(t)$ remains comparable to $E(t)$, with precise bounds

$$\Theta_1 E(t) \leq L(t) \leq \Theta_2 E(t), \quad \Theta_{1,2} = 1 \mp \varepsilon C_{\Upsilon},$$

where $C_{\Upsilon}$ is a computable constant determined by system parameters (Siriki et al., 20 Dec 2025).

5. Stability Results Enabled by Non-Standard Functionals

The main achievement of non-standard Lyapunov approaches is the proof of uniform exponential decay rates for highly non-uniform or complex PDEs. Via abstract semigroup theory and Lumer-Phillips theorem, m-dissipativity of the generator is established in the weighted space. Integration by parts and boundary term estimates yield a differential inequality of the form:

$\frac{d}{dt}L(t) \leq -\delta E(t),$

with boundary and delay contributions absorbed by the specific form of $G(t)$. Application of Komornik’s abstract decay theorems then delivers

$E(t) \leq C e^{-\omega t} E(0)$

with explicit bounds for $C$ and decay rate $\omega$ (Siriki et al., 20 Dec 2025).

A plausible implication is that non-standard Lyapunov functionals generalize multiplier techniques to settings with delay, distributed feedback, and operator degeneracy, which are prevalent in modern boundary control of flexible structures and networks.

6. Extensions and Related Methodologies

Non-standard Lyapunov constructions interface closely with:

• Multiplier and integrating-factor methods,
• Mikhaĭlov-type functionals for strong and weak damping,
• Delay-adapted transport variables in abstract boundary energy spaces,
• Operator-theoretic frameworks for systems with non-self-adjoint generators.

In degenerate, singular, or discontinuous settings, distributional methods (Dias et al., 2020) supply additional regularization and matching conditions, but Lyapunov-based analysis remains central for stability proofs rather than existence and uniqueness.

Current research continues to develop sharper forms of $G(t)$ adapted to fractional operators, inhomogeneous damping distributions, and networked boundary control, with the expectation of broadening the class of distributed systems amenable to rigorous exponential stabilization.

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Non-standard Lyapunov functionals thus constitute a foundational analytic tool for stabilization and energy decay in distributed systems subject to inhomogeneity, degeneracy, and temporal boundary complexity (Siriki et al., 20 Dec 2025, V. et al., 2015).