Composite Lyapunov Criteria for Stability and Convergence with Applications to Optimization Dynamics

Abstract

We propose a composite Lyapunov framework for nonlinear autonomous systems that ensures strict decay through a pair of differential inequalities. The approach yields integral estimates, quantitative convergence rates, vanishing of dissipation measures, convergence to a critical set, and semistability under mild conditions, without relying on invariance principles or compactness assumptions. The framework unifies convergence to points and sets and is illustrated through applications to inertial gradient systems and Primal–Dual gradient flows.

1. Introduction and Problem Setting

The analysis of asymptotic behavior in nonlinear dynamical systems remains a central theme in control theory and applied mathematics. Classical methods, such as Lyapunov’s direct approach, LaSalle’s invariance principle, and Matrosov-type criteria, form the foundation of stability analysis for many systems, including mechanical, optimization, and distributed models. Despite their effectiveness, these tools have structural limitations. LaSalle’s invariance principle identifies invariant sets but does not, by itself, guarantee convergence to them, while Matrosov-type conditions often provide only weak asymptotic results, usually expressed through the vanishing of dissipation measures in the limit inferior sense. Strong convergence typically requires additional compactness or stability assumptions on the invariant set.

The proposed approach is inspired by the classical development of asymptotic stability theory—from LaSalle’s invariance principle and Matrosov’s auxiliary–function method to later reinterpretations, such as the LaSalle-type formulation in [1]. In contrast to these methods, it employs a single composite Lyapunov function that achieves strict decay directly, without the use of nested or auxiliary functions. LaSalle’s invariance principle (see, e.g., [2]) provides a basic result: when a Lyapunov function is nonincreasing along trajectories, convergence is guaranteed only to the largest invariant set where its derivative vanishes. This result is qualitative and often depends on compactness.

To address these limitations, Matrosov [3] introduced an auxiliary function whose derivative remains strictly negative on the set where the derivative of the Lyapunov function vanishes, thereby establishing asymptotic stability without assuming compactness. This idea was later developed by Loria, Mazenc, and Teel [4], who proposed a nested Matrosov theorem to handle multiple auxiliary functions and nonautonomous systems, ensuring uniform convergence under persistency-of-excitation conditions. Further extensions were given by Mazenc and Nešić [5] for time-varying systems and by Teel et al. [6] for differential inclusions, where tailored Matrosov functions were used to prove attractivity of compact sets under arbitrary switching among finitely many vector fields. In the classical ODE setting, Astolfi and Praly [1] formulated a LaSalle-type version of Matrosov’s theorem, extending the analysis to systems with multiple equilibria and linking Matrosov’s reasoning with the invariance principle through a family of auxiliary functions and weak decay conditions.

This work develops a constructive alternative that derives strict decay directly from a pair of differential inequalities. Consider the autonomous system

$$\dot{x}\left(t\right)=f\left(x\left(t\right)\right),x\left(0\right)=x_0\in {\mathbb{R}}^n,$$

(1)

where $f:{\mathbb{R}}^n\to {\mathbb{R}}^n$ is locally Lipschitz. Under this assumption, the system admits a unique solution $x(·)$ on a maximal interval of existence. When needed, we assume the system is forward complete so that every trajectory is defined for all $t\ge 0$. The equilibrium set is

$$E_f:=\{x\in {\mathbb{R}}^n:f\left(x\right)=0\}.$$

A point $x^{\ast }\in E_f$ is Lyapunov stable if, for every neighborhood U of $x^{\ast }$, there exists a smaller neighborhood $V\subset U$ such that all trajectories starting in V remain in U for all future times. If, in addition, $x\left(t\right)\to x^{\ast }$ as $t\to \infty$, the equilibrium is asymptotically stable.

In this paper, the analysis extends beyond isolated equilibria to the stability of sets associated with system (1), with particular attention to pointwise asymptotic stability of a set (PAS). Pointwise asymptotic stability means that each point of the set is Lyapunov stable and that every trajectory starting near the set converges to some limit within it.

To clarify this notion, note that in the literature, PAS is typically defined with respect to the equilibrium set (see, e.g., [7,8]), while several works use the term semistability to describe the same property when restricted to equilibria (see, e.g., [9,10]). In this paper, the term PAS is used for an arbitrary set, whereas the term semistability is reserved for the case where the set coincides with the equilibrium set. An equilibrium is, therefore, semistable if it is Lyapunov stable and every trajectory starting near it converges to a (possibly different) Lyapunov stable equilibrium. When the equilibrium set reduces to a single point, semistability and asymptotic stability coincide.

The concepts of pointwise asymptotic stability and semistability provide a natural framework for systems with nonisolated equilibria and have been studied in various settings, including differential equations, differential inclusions, and hybrid systems (see, for instance, [7,8,9,10,11,12,13]). These works form the conceptual background for the results developed in this paper.

Having established the relevant stability notions, we now introduce the analytical framework used to verify them. The proposed approach constructs a composite Lyapunov function from two differential inequalities that capture the system’s dissipative behavior. Specifically, let $V_1,V_2:{\mathbb{R}}^n\to \mathbb{R}$ be continuously differentiable functions and $N_1,N_2:{\mathbb{R}}^n\to [0,\infty )$ be continuous, nonnegative functions. The functions $V_1$ and $V_2$ act as Lyapunov candidates, while $N_1$ and $N_2$ measure the dissipation reflected in their derivatives. Along the trajectories of (1), they satisfy

$${\dot{V}}_1\left(x\right)\le -N_1\left(x\right),{\dot{V}}_2\left(x\right)\le -N_2\left(x\right)+h\left(N_1\left(x\right)\right),$$

where $h:[0,\infty )\to [0,\infty )$ is continuous with $h\left(0\right)=0$.

The quantities $N_1$ and $N_2$ are called observables because they measure the rate of dissipation in the system and indicate how far the trajectory is from its asymptotic regime. They can be interpreted as measurable indicators of decay: each $N_i\left(x\right)$ is a nonnegative quantity that vanishes precisely when the corresponding Lyapunov function $V_i$ stops decreasing. In this sense, $N_i$ “observes” the energy dissipation at state x. Their role is twofold. Analytically, they quantify the dissipation appearing in the inequalities that govern the time derivatives of $V_1$ and $V_2$, providing a direct link to integral estimates and asymptotic convergence. Geometrically, their simultaneous vanishing defines the long-term behavior of the system. The set

$$E:=\{x\in {\mathbb{R}}^n:N_1\left(x\right)=0,N_2\left(x\right)=0\}$$

thus characterizes the asymptotic regime approached by trajectories and offers an observable-based representation of equilibria. This viewpoint is especially useful in nonsmooth or distributed settings where the vector field f may not be explicitly available, but the dissipation measures $N_1$ and $N_2$ remain well-defined.

Since $N_1$ and $N_2$ encode complementary information about system dissipation, it is natural to combine them through a single Lyapunov expression that reflects their joint effect. The first function, $V_1$, monitors the main dissipation quantified by $N_1$, while $V_2$ remains sensitive when ${\dot{V}}_1$ vanishes. By coupling them through the gain $h\left(N_1\right)$, the composite function $W=V_1+\delta V_2$ achieves strict decay for a suitable $\delta >0$, providing a unified Lyapunov framework that ensures convergence to the set E.

A small-gain condition on h ensures that the influence of $N_1$ on the decay of $V_2$ does not cancel the overall dissipation. To capture their combined effect, we introduce the composite Lyapunov function

$$W\left(x\right)=V_1\left(x\right)+\delta V_2\left(x\right),\delta >0,$$

and show that, for a suitable $\delta$, there exists $\gamma >0$ such that

$$\dot{W}\left(x\right)\le -\gamma \left(N_1\left(x\right)+N_2\left(x\right)\right).$$

This strict-decay inequality combines the information from both differential inequalities, enforcing the vanishing of $N_1\left(x\left(t\right)\right)$ and $N_2\left(x\left(t\right)\right)$ along every trajectory. It forms the cornerstone of the analysis developed in this paper and shows that the proposed approach achieves a direct decay relation, without relying on hierarchical auxiliary–function constructions or successive negativity checks.

Every bounded trajectory satisfies $\mathrm{dist}\left(x\right(t),E)\to 0$ as $t\to \infty$, meaning that trajectories asymptotically approach the critical set where both dissipation measures vanish. Since equilibria make $N_1$ and $N_2$ equal to zero, the equilibrium set $E_f$ is contained in E. In many cases, the two sets coincide; therefore, trajectories approach the equilibrium set, leading to semistability when each equilibrium is Lyapunov stable. More generally, if every point of E is Lyapunov stable, then E is pointwise asymptotically stable (PAS). The strict-decay inequality further provides integral estimates and convergence of observables, which form the basis for these stability results. Finally, when E reduces to a single equilibrium, global asymptotic stability follows.

This composite Lyapunov construction unifies and extends the classical Lyapunov and Matrosov approaches. It establishes convergence and stability properties without relying on compactness or invariance principles and applies naturally to nonlinear and optimization-driven systems. In particular, it encompasses the dynamics of inertial gradient-like systems and Primal–Dual gradient flows that arise in constrained optimization and networked control problems. These two classes of systems serve as detailed case studies in the sequel, illustrating how the general results of the paper guarantee convergence and stability of their equilibrium sets. Beyond optimization, the same framework can also model dynamic adjustment processes in economics, such as signaling-based Cournot competition [14], where the evolution of agents’ strategies toward equilibrium can be analyzed through stability and convergence properties.

In addition to qualitative convergence, the approach provides quantitative estimates for the rate at which trajectories approach the critical set, linking decay inequalities with integral estimates and error–bound conditions. These results make it possible to assess how rapidly the system stabilizes, offering explicit performance guarantees that connect Lyapunov decay with convergence speed—an aspect of practical relevance in optimization, control, and economic dynamics.

2. Main Results

The following results establish the complete framework built on the pair of differential inequalities introduced earlier. The first step (Section 2.1) proves that a suitable combination of $V_1$ and $V_2$ yields a composite Lyapunov function that decreases strictly along trajectories. Subsequent subsections derive their main consequences—integral bounds, convergence to the critical set, quantitative decay rates, and stability properties. Together, they provide a unified and constructive approach to analyzing asymptotic behavior without relying on invariance principles or compactness assumptions.

2.1. Strict Decay of the Composite Function

In the classical Matrosov-type theorem, the auxiliary Lyapunov functions $V_i$ are assumed to be continuously differentiable, and their derivatives ${\dot{V}}_i$ must satisfy a uniform continuity condition along trajectories to ensure strict decay via integral arguments. In contrast, the present framework removes these smoothness and global regularity assumptions. The quantities $N_i$ are only required to be nonnegative, locally bounded, and continuous (without requiring uniform continuity), which is sufficient to establish the strict-decay inequality for the composite function $W=V_1+\delta V_2$. This highlights the analytical simplicity and wider applicability of the proposed criterion while retaining the core coupling idea behind Matrosov’s construction.

We begin with the fundamental decay result that underlies the entire framework.

Theorem 1

(Strict Decay of a Composite Lyapunov Function). Suppose there exist continuously differentiable functions $V_1,V_2:{\mathbb{R}}^n\to \mathbb{R}$, continuous nonnegative functions $N_1,N_2:{\mathbb{R}}^n\to [0,+\infty )$, and a continuous function $h:[0,+\infty )\to [0,+\infty )$ with $h\left(0\right)=0$ such that, along every solution,

$${\dot{V}}_1\left(x\right)\le -N_1\left(x\right),{\dot{V}}_2\left(x\right)\le -N_2\left(x\right)+h\left(N_1\left(x\right)\right).$$

Assume in addition that

$$L:=\underset{r>0}{sup}{\displaystyle \frac{h\left(r\right)}{r}}<+\infty .$$

Then, for any $\delta \in (0,1/L)$ and $W\left(x\right):=V_1\left(x\right)+\delta V_2\left(x\right)$, there exists a constant

$$\gamma :=min\{1-\delta L,\delta \}>0$$

such that, along all solutions,

$$\dot{W}\left(x\right)\le -\gamma \left(N_1\left(x\right)+N_2\left(x\right)\right).$$

Proof. 

Take any solution $x(·)$ of the system. By assumption, the functions $V_1$ and $V_2$ satisfy

$${\dot{V}}_1\left(x\right)\le -N_1\left(x\right),{\dot{V}}_2\left(x\right)\le -N_2\left(x\right)+h\left(N_1\left(x\right)\right).$$

Fix a parameter $\delta >0$ and define the composite Lyapunov function

$$W\left(x\right):=V_1\left(x\right)+\delta V_2\left(x\right).$$

Differentiating along the trajectory gives

$$\dot{W}\left(x\right)={\dot{V}}_1\left(x\right)+\delta {\dot{V}}_2\left(x\right)\le -N_1\left(x\right)-\delta N_2\left(x\right)+\delta h\left(N_1\left(x\right)\right).$$

Now apply the global slope bound. By definition of

$$L:=\underset{r>0}{sup}{\displaystyle \frac{h\left(r\right)}{r}},$$

we have $h\left(s\right)\le Ls$ for every $s\ge 0$. In particular, for $s=N_1\left(x\right)$,

$$\delta h\left(N_1\left(x\right)\right)\le \delta L{N}_1\left(x\right).$$

Substituting this into the inequality for $\dot{W}$ yields

$$\dot{W}\left(x\right)\le -(1-\delta L)N_1\left(x\right)-\delta N_2\left(x\right).$$

Choose any $\delta \in (0,1/L)$. Then both coefficients $1-\delta L$ and $\delta$ are positive. Since $N_1\left(x\right),N_2\left(x\right)\ge 0$, we can estimate the right-hand side by pulling out the smaller of the two coefficients:

$$(1-\delta L)N_1\left(x\right)+\delta N_2\left(x\right)\ge min\{1-\delta L,\delta \}\left(N_1\left(x\right)+N_2\left(x\right)\right).$$

Therefore,

$$\dot{W}\left(x\right)\le -\gamma \left(N_1\left(x\right)+N_2\left(x\right)\right),\gamma :=min\{1-\delta L,\delta \}>0.$$

This proves that the composite Lyapunov function decreases strictly along every solution, with a decay rate controlled by $\gamma$. □

The constants in the strict-decay inequality can, in fact, be optimized, leading to the following refinement.

Remark 1

(Role of h and analytical form of constants). The function $h:[0,\infty )\to [0,\infty )$ represents the coupling gain that links the decay channels $N_1$ and $N_2$. It measures how the dissipation detected by $N_1$ influences the derivative of $V_2$ through the inequality ${\dot{V}}_2\le -N_2+h\left(N_1\right)$. The continuity of h with $h\left(0\right)=0$ ensures compatibility of the two inequalities, while the bounded-slope condition $h\left(r\right)\le Lr$ (i.e., $h\left(r\right)/r<\infty$ for $r>0$) is a small-gain assumption guaranteeing that the feedback of $N_1$ into ${\dot{V}}_2$ cannot dominate the total decay. Under this bound, we obtain the explicit computation

$$\dot{W}={\dot{V}}_1+\delta {\dot{V}}_2\le -(1-\delta L)N_1-\delta N_2,$$

which yields a strict-decay inequality $\dot{W}\le -\gamma (N_1+N_2)$ for any $\delta \in (0,1/L)$ with $\gamma =min\{1-\delta L,\delta \}>0$. Hence, the constants $(\delta ,\gamma )$ can be chosen uniformly in compact form, providing a clear analytical interpretation of the parameters that appear in Theorem 1. In the next remark, the optimal choice of $(\delta ,\gamma )$ is characterized analytically. All constants can be written in compact analytical form through the single parameter pair $(L,\delta )$, yielding $\gamma =min\{1-\delta L,\delta \}$.

Remark 2

(Optimal choice of constants). The decay estimate in Theorem 1 holds for any $\delta \in (0,1/L)$ with

$$\dot{W}\left(x\right)\le -\gamma \left(N_1\left(x\right)+N_2\left(x\right)\right),\gamma =min\{1-\delta L,\delta \}.$$

The guaranteed rate γ depends on the tuning parameter δ. Since $1-\delta L$ decreases linearly with δ while δ increases linearly, the quantity γ is maximized when the two terms coincide, i.e.,

$$1-\delta L=\delta .$$

Solving for δ gives the optimal choice

$${\delta }^{\ast }={\displaystyle \frac{1}{1+L}}.$$

Substituting this value into the expression for γ yields

$${\gamma }^{\ast }={\displaystyle \frac{1}{1+L}}.$$

Thus, the sharpest form of the strict-decay inequality is

$$\dot{W}\left(x\right)\le -{\displaystyle \frac{1}{1+L}}\left(N_1\left(x\right)+N_2\left(x\right)\right).$$

2.2. Local Strict Decay Under Bounded Trajectories

The global result in Theorem 1 relies on the uniform slope bound $L={sup}_{r>0}h\left(r\right)/r<\infty$, which guarantees a global decay rate valid for all trajectories. When such a bound is available only on a bounded region or along a specific trajectory, a local version can be established with trajectory-dependent constants. The constant L in Theorem 1 controls the interaction term $h\left(N_1\right)$ globally. In many systems, however, $h\left(r\right)/r$ may be bounded only on the range of $N_1$ actually visited by the trajectory. To capture this situation, we restrict attention to a positively invariant set $\Omega \subseteq {\mathbb{R}}^n$ that contains the entire trajectory $\left\{x\right(t):t\ge 0\}$. On such a set, the ratio $h\left(s\right)/s$ remains finite, allowing a local version of the strict-decay inequality with constants that depend on $\Omega$ and ultimately on the initial condition.

Theorem 2

(Local Strict Decay with Optimal Constants). Suppose there exist $V_1,V_2\in C^1\left({\mathbb{R}}^n\right)$, continuous functions $N_1,N_2:{\mathbb{R}}^n\to [0,\infty )$, and a continuous function $h:[0,\infty )\to [0,\infty )$ with $h\left(0\right)=0$ such that, along every trajectory,

$${\dot{V}}_1\left(x\right)\le -N_1\left(x\right),{\dot{V}}_2\left(x\right)\le -N_2\left(x\right)+h\left(N_1\left(x\right)\right).$$

Let $\Omega \subseteq {\mathbb{R}}^n$ be any positively invariant set containing the trajectory $\left\{x\right(t):t\ge 0\}$ (for instance, $\Omega ={\mathbb{R}}^n$ for a global statement). Define

$$S_{\Omega }:=\{N_1\left(x\right):x\in \Omega \},B_{\Omega }:=\underset{s\in S_{\Omega }\setminus \left\{0\right\}}{sup}{\displaystyle \frac{h\left(s\right)}{s}}\in [0,\infty ].$$

If $B_{\Omega }<\infty$, then for any $\delta \in (0,1/B_{\Omega })$ the composite function $W:=V_1+\delta V_2$ satisfies, along the trajectory,

$$\dot{W}\left(x\right)\le -\gamma \left(N_1\left(x\right)+N_2\left(x\right)\right),\gamma :=min\{1-\delta B_{\Omega },\delta \}>0.$$

Moreover, the decay rate is optimized by choosing

$${\delta }^{★}={\displaystyle \frac{1}{1+B_{\Omega }}},{\gamma }^{★}={\displaystyle \frac{1}{1+B_{\Omega }}},$$

which yields the explicit bound

$$\dot{W}\left(x\right)\le -{\displaystyle \frac{1}{1+B_{\Omega }}}\left(N_1\left(x\right)+N_2\left(x\right)\right)along the trajectory.$$

Sketch of Proof.

The argument follows exactly as in Theorem 1: starting from $\dot{W}={\dot{V}}_1+\delta {\dot{V}}_2\le -N_1-\delta N_2+\delta h\left(N_1\right)$ and using the local bound $h\left(s\right)\le B_{\Omega }s$ for $s\in S_{\Omega }$ gives

$$\dot{W}\le -(1-\delta B_{\Omega })N_1-\delta N_2.$$

Choosing $\delta \in (0,1/B_{\Omega })$ yields $\dot{W}\le -\gamma (N_1+N_2)$ with $\gamma =min\{1-\delta B_{\Omega },\delta \}>0$, and optimizing over $\delta$ produces ${\delta }^{★}={\gamma }^{★}=1/(1+B_{\Omega })$. □

Corollary 1

(Bounded-Trajectory Version with $L_R$). If, along a trajectory $x(·)$, one has $N_1\left(x\left(t\right)\right)\le R$ for some $R>0$, define

$$L_R:=\underset{0\le s\le R}{sup}{\displaystyle \frac{h\left(s\right)}{s}}<\infty .$$

Then, for any $\delta \in (0,1/L_R)$, the composite function $W=V_1+\delta V_2$ satisfies

$$\dot{W}\left(x\right)\le -\gamma (N_1\left(x\right)+N_2\left(x\right)),\gamma =min\{1-\delta L_R,\delta \}>0,$$

along that trajectory. The optimal choice is

$${\delta }^{★}={\displaystyle \frac{1}{1+L_R}},{\gamma }^{★}={\displaystyle \frac{1}{1+L_R}}.$$

Remark 3

(Useful Specializations).

• Global version: If $L:={sup}_{r>0}h\left(r\right)/r<\infty$, take $\Omega ={\mathbb{R}}^n$, so that $B_{\Omega }=L$.
• Bounded-trajectory version: If the solution is bounded in a positively invariant K, set $\Omega =K$; then $B_{\Omega }={sup}_{0\le s\le R_K}h\left(s\right)/s$ with $R_K:={sup}_{x\in K}{N}_1\left(x\right)$.

2.3. Integral Estimates and Vanishing of Observables

The next result translates the strict-decay inequality into quantitative information along trajectories. It shows that the total dissipation of the observables is finite and that, under a mild continuity assumption, both observables vanish asymptotically. This step connects the decay property of W to the convergence behavior of the system.

Proposition 1

(Integral estimates and convergence of observables). Under the assumptions of Theorem 1, the composite function $W\left(x\right(t\left)\right)$ is nonincreasing along solutions and, therefore, converges to a finite limit as $t\to \infty$. Moreover,

$${\int }_0^{\infty }\left(N_1\left(x\left(t\right)\right)+N_2\left(x\left(t\right)\right)\right)dt<\infty .$$

If, in addition, the functions $t\mapsto N_i\left(x\left(t\right)\right)$ are uniformly continuous on $[0,\infty )$, then the finiteness of the above integral implies

$$\underset{t\to \infty }{lim}{N}_1\left(x\left(t\right)\right)=0,\underset{t\to \infty }{lim}{N}_2\left(x\left(t\right)\right)=0.$$

Proof. 

By Theorem 1 there exists $\gamma >0$ such that, along every solution $x(·)$,

$$\dot{W}\left(x\left(t\right)\right)\le -\gamma \left(N_1\left(x\left(t\right)\right)+N_2\left(x\left(t\right)\right)\right)\mathrm{for}\mathrm{all}t\ge 0.$$

(2)

Since the right–hand side is nonpositive, the map $t\mapsto W\left(x\right(t\left)\right)$ is nonincreasing. In particular, for all $T\ge 0$,

$$W\left(x\right(T\left)\right)\le W\left(x\right(0\left)\right).$$

Integrating (2) on $[0,T]$ gives

$$W\left(x\left(T\right)\right)-W\left(x\left(0\right)\right)\le -\gamma {\int }_0^T\left(N_1\left(x\left(t\right)\right)+N_2\left(x\left(t\right)\right)\right)dt,$$

hence,

$${\int }_0^T\left(N_1\left(x\left(t\right)\right)+N_2\left(x\left(t\right)\right)\right)dt\le {\displaystyle \frac{W\left(x\right(0\left)\right)-W\left(x\right(T\left)\right)}{\gamma }}.$$

(3)

Because W is nonincreasing along the trajectory, the limit ${\displaystyle \underset{T\to \infty }{lim}W\left(x\left(T\right)\right)}$ exists in $[-\infty ,+\infty )$. In Lyapunov constructions, one typically chooses $V_1,V_2\ge 0$, in which case $W\ge 0$ and the limit is finite. Under this mild bounded–below condition, passing to the limit $T\to \infty$ in (3) yields

$${\int }_0^{\infty }\left(N_1\left(x\left(t\right)\right)+N_2\left(x\left(t\right)\right)\right)dt\le {\displaystyle \frac{W\left(x\left(0\right)\right)-{lim}_{T\to \infty }W\left(x\left(T\right)\right)}{\gamma }}<\infty .$$

We now prove the asymptotic vanishing of each observable under the uniform continuity assumption. Fix $i\in \{1,2\}$ and define $g\left(t\right):=N_i\left(x\left(t\right)\right)$. Then $g:[0,\infty )\to [0,\infty )$ is uniformly continuous by hypothesis, and ${\displaystyle {\int }_0^{\infty }g\left(t\right)dt<\infty }$ by the previous estimate. We claim that ${\displaystyle \underset{t\to \infty }{lim}g\left(t\right)=0}$. Suppose, for contradiction, that this limit does not exist or is not zero. Then there exists $\epsilon >0$ and a sequence of times $t_k\to \infty$ such that $g\left(t_k\right)\ge \epsilon$ for all k. By uniform continuity of g, there exists $\delta \in (0,1)$ such that

$$|t-s|<\delta \Rightarrow \left|g\right(t)-g(s\left)\right|<\epsilon /2.$$

In particular, $g\left(t\right)\ge \epsilon /2$ for all $t\in [t_k-\delta /2,t_k+\delta /2]$. By taking a subsequence (still denoted $t_k$) with pairwise distance at least $\delta$, these intervals are disjoint. Therefore,

$${\int }_0^{\infty }g\left(t\right)dt\ge \sum _{k=1}^{\infty }{\int }_{t_k-\delta /2}^{t_k+\delta /2}g\left(t\right)dt\ge \sum _{k=1}^{\infty }{\displaystyle \frac{\epsilon }{2}}\delta =+\infty ,$$

which contradicts ${\displaystyle {\int }_0^{\infty }g\left(t\right)dt<\infty }$. Hence, ${\displaystyle \underset{t\to \infty }{lim}g\left(t\right)=0}$. Since $i\in \{1,2\}$ was arbitrary, we conclude

$$\underset{t\to \infty }{lim}{N}_1\left(x\left(t\right)\right)=\underset{t\to \infty }{lim}{N}_2\left(x\left(t\right)\right)=0.$$

Finally, combining the nonincreasing property of W with its bounded–below property along the trajectory (e.g., $V_1,V_2\ge 0$) shows that $W\left(x\right(t\left)\right)$ converges to a finite limit as $t\to \infty$. This completes the proof. □

Remark 4.

Classical Matrosov-type theorems, including the formulation proposed in [1], typically establish only the weaker conclusion

$$\underset{t\to \infty }{lim\; inf}{N}_i\left(x\left(t\right)\right)=0,$$

which guarantees that the observables vanish along subsequences but does not ensure the convergence. In contrast, the strict-decay framework developed here, together with the mild uniform continuity assumption along trajectories, yields the full limits

$$\underset{t\to \infty }{lim}{N}_1\left(x\left(t\right)\right)=\underset{t\to \infty }{lim}{N}_2\left(x\left(t\right)\right)=0.$$

This provides a strictly stronger asymptotic conclusion than those available in the classical Matrosov setting.

Bounded trajectories and local Lipschitz continuity of the dynamics f and the observables $N_i$ imply that $t\mapsto N_i\left(x\left(t\right)\right)$ is uniformly continuous on $[0,\infty )$. Hence, by Proposition 1, $N_i\left(x\left(t\right)\right)\to 0$ as $t\to \infty$.

2.4. Convergence to the Critical Set

The vanishing of the observables characterizes the asymptotic behavior of trajectories in terms of the critical set

$$E:=\{x\in {\mathbb{R}}^n:N_1\left(x\right)=0,N_2\left(x\right)=0\}.$$

Notice that the critical set E may, in general, strictly contain the equilibrium set $E_f:=\{x:f\left(x\right)=0\}$. This happens when the observables $N_i$ capture only partial dissipation effects that can vanish without forcing $f\left(x\right)=0$. In such cases, trajectories approach the largest invariant subset of E, and convergence to equilibria in $E_f$ requires additional structural or invariance conditions on the dynamics.

The following result shows that the vanishing of the observables forces every bounded trajectory to approach this set asymptotically.

Theorem 3

(Convergence to E). Suppose that, along a trajectory $x(·)$, the functions $t\mapsto N_i\left(x\left(t\right)\right)$ are uniformly continuous on $[0,\infty )$ for $i=1,2$. Then

$$\mathrm{dist}\left(x\right(t),E)\to 0ast\to \infty .$$

Proof. 

Let $M\left(x\right):=N_1\left(x\right)+N_2\left(x\right)$, so that $E=\{x:M(x)=0\}$. By Proposition 1 and the uniform continuity of $t\mapsto N_i\left(x\left(t\right)\right)$ on $[0,\infty )$, we have

$$M\left(x\right(t\left)\right)\to 0\mathrm{as}t\to \infty .$$

Fix $\epsilon >0$ and consider the closed set

$$F_{\epsilon }:=\{x\in {\mathbb{R}}^n:dist(x,E)\ge \epsilon \}.$$

On $F_{\epsilon }$ the function M cannot vanish, since points of $F_{\epsilon }$ are at least $\epsilon$ away from E. By the continuity of M, this implies that on every compact subset of $F_{\epsilon }$, the function M admits a strictly positive minimum. Since the trajectory is bounded, its image $\overline{\left\{x\right(t):t\ge 0\}}$ is compact; hence, this compactness condition is automatically satisfied for the portion of the trajectory lying in $F_{\epsilon }$. In particular, along the bounded portion of the trajectory that may lie in $F_{\epsilon }$, there exists $\eta \left(\epsilon \right)>0$ such that

$$dist\left(x\right(t),E)\ge \epsilon \Rightarrow M\left(x\right(t\left)\right)\ge \eta \left(\epsilon \right).$$

Suppose, toward a contradiction, that there exists a sequence $t_k\to \infty$ with $dist(x\left(t_k\right),E)\ge \epsilon$ for all k. Then by the implication above,

$$M\left(x\left(t_k\right)\right)\ge \eta \left(\epsilon \right)\mathrm{for}\mathrm{all}k.$$

But this contradicts the fact that $M\left(x\right(t\left)\right)\to 0$ as $t\to \infty$.

Therefore, such a sequence cannot exist, and we conclude

$$dist\left(x\right(t),E)\to 0\mathrm{as}t\to \infty ,$$

which proves the claim. □

The convergence stated in Theorem 3 applies not only to isolated equilibria but also when the critical set E forms a continuum, such as a curve or manifold of equilibria. In this case, the result ensures an asymptotic approach to the set E in the sense of semistability, rather than convergence to a single equilibrium point.

Remark 5

(Separation property). The argument uses that $M\left(x\right)>0$ whenever $dist(x,E)\ge \epsilon$. This separation property—a standard local error–bound condition for the zero set of a nonnegative continuous function—holds automatically on compact subsets or whenever M has no flat valleys away from E. It ensures that the implication $M\left(x\right(t\left)\right)\to 0\Rightarrow dist\left(x\right(t),E)\to 0$ is mathematically sound without assuming global boundedness.

Building on Theorem 3, which ensures convergence of trajectories to the critical set E, we obtain the following consequence.

2.5. Quantitative Convergence Rate

This subsection strengthens the qualitative convergence result by giving explicit rates under two simple hypotheses: strict decay of W and a local error bound linking the observables to the distance from E. Integrating the decay yields an $L^2(0,\infty )$ estimate for the distance. An averaging step on each window $[T,2T]$ then produces times $\tau \in [T,2T]$ with $\mathrm{dist}(x\left(\tau \right),E)=O\left(T^{-1/2}\right)$. If the distance is eventually nonincreasing, the same idea on the backward window $[t/2,t]$ gives a pointwise rate $\mathrm{dist}(x\left(t\right),E)=O\left(t^{-1/2}\right)$ for large t. A closing remark records that a local quadratic growth of W near E upgrades these sublinear bounds to exponential decay, with constants explicit in terms of the decay modulus $\gamma$ and the error–bound parameters.

Proposition 2

(Quantitative convergence from a local error bound). Assume that along a trajectory $x(·)$, the strict-decay inequality holds:

$$\dot{W}\left(t\right)\le -\gamma [N_1\left(x\left(t\right)\right)+N_2\left(x\left(t\right)\right)],\gamma >0.$$

Suppose there exists a neighborhood U of E and constants $c_1,c_2>0$ such that

$$N_i\left(x\right)\ge c_i\mathrm{dist}{(x,E)}^{2}forallx\in U,i=1,2.$$

Set $c:=c_1+c_2$. Then:

(1)

${\mathit{L}}^{\mathbf{2}}$–integrability.  Once the trajectory enters U,

$${\int }_0^{+\infty }\mathrm{dist}{(x\left(t\right),E)}^{2}dt\le {\displaystyle \frac{W\left(x\left(0\right)\right)-{lim}_{t\to \infty }W\left(x\left(t\right)\right)}{\gamma c}}<+\infty .$$

(2)

Subsequence rate.  Define

$$K:=\sqrt{{\displaystyle \frac{W\left(x\left(0\right)\right)-{lim}_{t\to \infty }W\left(x\left(t\right)\right)}{\gamma c}}}.$$

Then for every $T>0$,

$${\int }_T^{2T}\mathrm{dist}{(x\left(t\right),E)}^{2}dt\le K^2,$$

hence, by averaging on $[T,2T]$, there exists $\tau \in [T,2T]$ with

$$\mathrm{dist}{(x\left(\tau \right),E)}^2\le {\displaystyle \frac{1}{T}}{\int }_T^{2T}\mathrm{dist}{(x\left(t\right),E)}^{2}dt\le {\displaystyle \frac{K^2}{T}},$$

so ${\displaystyle \mathrm{dist}(x\left(\tau \right),E)\le \frac{K}{\sqrt{T}}}$.

(3)

Pointwise rate.  If, in addition, $t\mapsto \mathrm{dist}\left(x\right(t),E)$ is uniformly continuous on $[0,+\infty )$ and eventually nonincreasing, then there exist $T_0>0$ and $C>0$ such that

$$\mathrm{dist}(x\left(t\right),E)\le {\displaystyle \frac{C}{\sqrt{t}}},\forall t\ge T_0.$$

Proof. 

Let $x(·)$ be a trajectory for which the strict-decay estimate

$$\dot{W}\left(t\right)\le -\gamma [N_1\left(x\left(t\right)\right)+N_2\left(x\left(t\right)\right)],\gamma >0,$$

(4)

holds. Assume there exist a neighborhood U of E and $c_1,c_2>0$ such that

$$N_i\left(x\right)\ge c_i\mathrm{dist}{(x,E)}^2\mathrm{for}\mathrm{all}x\in U,i=1,2,$$

(5)

and set $c:=c_1+c_2$.

Fix any time $T_U\ge 0$ such that $x\left(t\right)\in U$ for all $t\ge T_U$ (for example, any $T_U$ after which the trajectory remains in U). All estimates below are written on $[T_U,\infty )$.

(1) 

${\mathit{L}}^{\mathbf{2}}$–integrability. On U, summing (5) gives

$$N_1\left(x\left(t\right)\right)+N_2\left(x\left(t\right)\right)\ge c\mathrm{dist}{(x\left(t\right),E)}^2(t\ge T_U).$$

Combining with (4) yields

$$\dot{W}\left(t\right)\le -\gamma c\mathrm{dist}{(x\left(t\right),E)}^2(t\ge T_U).$$

(6)

Integrate (6) on $[T_U,T]$:

$$\gamma c{\int }_{T_U}^T\mathrm{dist}{(x\left(t\right),E)}^{2}dt\le W\left(x\left(T_U\right)\right)-W\left(x\left(T\right)\right).$$

Let $T\to \infty$; since $W\left(x\right(t\left)\right)$ is nonincreasing and $W\ge 0$, ${lim}_{t\to \infty }W\left(x\left(t\right)\right)$ exists and is finite. Hence,

$${\int }_{T_U}^{\infty }\mathrm{dist}{(x\left(t\right),E)}^{2}dt\le {\displaystyle \frac{W\left(x\left(T_U\right)\right)-{lim}_{t\to \infty }W\left(x\left(t\right)\right)}{\gamma c}}\le {\displaystyle \frac{W\left(x\left(0\right)\right)-{lim}_{t\to \infty }W\left(x\left(t\right)\right)}{\gamma c}}<\infty .$$

Since the integrand is nonnegative, extending the lower limit to 0 preserves finiteness, proving (1).

(2) 

Subsequence $\mathit{O}\left({\mathit{T}}^{-\mathbf{1}/\mathbf{2}}\right)$ rate. For any $T\ge T_U$, integrate (6) on $[T,2T]$:

$$\gamma c{\int }_T^{2T}\mathrm{dist}{(x\left(t\right),E)}^{2}dt\le W\left(x\left(T\right)\right)-W\left(x\left(2T\right)\right)\le W\left(x\left(0\right)\right)-W_{\infty },$$

where $W_{\infty }:={lim}_{t\to \infty }W\left(x\left(t\right)\right)$ exists since $W\ge 0$ and is nonincreasing. Define

$$K:=\sqrt{{\displaystyle \frac{W\left(x\left(0\right)\right)-W_{\infty }}{\gamma c}}},\mathrm{so}{\int }_T^{2T}\mathrm{dist}{(x\left(t\right),E)}^{2}dt\le K^2.$$

Because $\left|\right[T,2T\left]\right|=T$, the average of $\mathrm{dist}{(x\left(t\right),E)}^2$ on $[T,2T]$ satisfies

$${\displaystyle \frac{1}{T}}{\int }_T^{2T}\mathrm{dist}{(x\left(t\right),E)}^{2}dt\le {\displaystyle \frac{K^2}{T}}.$$

If $\mathrm{dist}{(x\left(t\right),E)}^2$ were strictly larger than this average for every $t\in [T,2T]$, integrating would give

$${\int }_T^{2T}\mathrm{dist}{(x\left(t\right),E)}^{2}dt>T·{\displaystyle \frac{1}{T}}{\int }_T^{2T}\mathrm{dist}{(x\left(s\right),E)}^{2}ds=K^2,$$

a contradiction. Hence, there exists $\tau \in [T,2T]$ such that

$$\mathrm{dist}{(x\left(\tau \right),E)}^2\le {\displaystyle \frac{1}{T}}{\int }_T^{2T}\mathrm{dist}{(x\left(t\right),E)}^{2}dt\le {\displaystyle \frac{K^2}{T}},$$

and therefore $\mathrm{dist}(x\left(\tau \right),E)\le {\displaystyle \frac{K}{\sqrt{T}}}$.

(3) 

Pointwise $\mathit{O}\left({\mathit{t}}^{-\mathbf{1}/\mathbf{2}}\right)$ under eventual monotonicity. Assume there exists $t_1\ge 0$ such that $\mathrm{dist}\left(x\right(t),E)$ is nonincreasing on $[t_1,\infty )$. For any $t\ge max\{2T_U,2t_1\}$, integrating (6) on $[t/2,t]$ gives

$${\int }_{t/2}^t\mathrm{dist}{(x\left(s\right),E)}^{2}ds\le K^2.$$

Since $\left|\right[t/2,t\left]\right|=t/2$, there exists $\tau \in [t/2,t]$ with

$$\mathrm{dist}{(x\left(\tau \right),E)}^2\le {\displaystyle \frac{2K^2}{t}}\Rightarrow \mathrm{dist}(x\left(\tau \right),E)\le {\displaystyle \frac{\sqrt{2}K}{\sqrt{t}}}.$$

By monotonicity and $\tau \le t$,

$$\mathrm{dist}(x\left(t\right),E)\le \mathrm{dist}(x\left(\tau \right),E)\le {\displaystyle \frac{\sqrt{2}K}{\sqrt{t}}}.$$

Thus, the claim holds with $T_0:=max\{2T_U,2t_1\}$ and $C:=\sqrt{2}K$.

The three claims complete the proof. □

Remark 6

(Analytical meaning of the condition (5)). The inequality (5) expresses a local error–bound or quadratic growth property that connects the observables $N_i$ to the distance from the equilibrium set E. It guarantees that, in a neighborhood of E, the dissipation terms $N_i\left(x\right)$ are sufficiently strong to dominate the local behavior of the trajectories, so that the strict decay of W translates into a quantitative decrease of $\mathrm{dist}\left(x\right(t),E)$. In this sense, (5) provides a clear analytical link between the decay of the Lyapunov function and the convergence rate. The condition formalizes the idea that $N_i\left(x\right)$ quantify the proximity to equilibrium with quadratic accuracy, which is essential for obtaining the explicit $O\left(t^{-1/2}\right)$ and exponential convergence rates stated in Proposition 2.

Remark 7.

If in Proposition 2, we only assume uniform continuity of $t\mapsto \mathrm{dist}\left(x\right(t),E)$, then for every large T there exist $\tau \in [T,2T]$ and a fixed $\Delta >0$ (independent of T) such that $\mathrm{dist}(x\left(s\right),E)\le K/\sqrt{T}+\epsilon$ for all $s\in [\tau ,\tau +\Delta ]$. This yields recurring short-interval bounds but not a global pointwise $O\left(t^{-1/2}\right)$ rate.

Remark 8

(Quadratic growth ⇒ exponential rate). Assume there exist $m>0$ and $r>0$ such that

$$W\left(x\right)-W_{\infty }\ge m\mathrm{dist}{(x,E)}^{2}whenever\mathrm{dist}(x,E)\le r.$$

(7)

Pick $t_0\ge 0$ large enough so that $\mathrm{dist}\left(x\right(t),E)\le r$ for all $t\ge t_0$ (which holds since $\mathrm{dist}\left(x\right(t),E)\to 0$). Then, for all $t\ge t_0$,

$$\dot{W}\left(t\right)\le -\gamma [N_1\left(x\left(t\right)\right)+N_2\left(x\left(t\right)\right)]\le -{\displaystyle \frac{\gamma c}{m}}[W\left(x\left(t\right)\right)-W_{\infty }].$$

By Grönwall inequality on $[t_0,+\infty )$,

$$W\left(x\left(t\right)\right)-W_{\infty }\le \left(W\left(x\left(t_0\right)\right)-W_{\infty }\right)e^{-(\gamma c/m)(t-t_0)}.$$

Using (7) again, we have

$$\mathrm{dist}(x\left(t\right),E)\le \sqrt{{\displaystyle \frac{W\left(x\left(t\right)\right)-W_{\infty }}{m}}}\le \sqrt{{\displaystyle \frac{W\left(x\left(t_0\right)\right)-W_{\infty }}{m}}}{e}^{-(\gamma c/\left(2m\right))(t-t_0)}.$$

The condition $W-W_{\infty }\ge m{\mathrm{dist}}^2$ is a local quadratic growth (error–bound) near E. It makes the energy gap equivalent to the squared distance; thus, strict decay yields exponential convergence of both W and $\mathrm{dist}(·,E)$. This assumption is standard and easily verified when W (or its potential Φ) satisfies a Polyak–Łojasiewicz inequality with a Lipschitz gradient. This condition implies the local quadratic growth (7) and, therefore, guarantees the exponential convergence established above.

Example 1

(exponential rate under PL). Consider the gradient flow $\dot{x}=-\nabla \Phi \left(x\right)$ with $\Phi \left(x\right)=\frac{k}{2}{\parallel x\parallel }^2$ ($k>0$). Then $W=\Phi$, $N_1\equiv 0$, $N_2={\parallel \nabla \Phi \left(x\right)\parallel }^2=k^2{\parallel x\parallel }^2$; hence, $\dot{W}=-N_2=-k^2{\parallel x\parallel }^2$. Since Φ satisfies the Polyak–Łojasiewicz (PL) inequality ${\parallel \nabla \Phi \left(x\right)\parallel }^2\ge 2k[\Phi \left(x\right)-\Phi \left(0\right)]$, we obtain the local quadratic growth $W-W_{\infty }\ge \frac{1}{k}{\parallel \nabla \Phi \left(x\right)\parallel }^2$, which matches (7). Therefore,

$$W\left(x\left(t\right)\right)-W_{\infty }\le \left(W\left(x\left(0\right)\right)-W_{\infty }\right)e^{-2kt}\mathrm{and}\mathrm{dist}(x\left(t\right),\left\{0\right\})=\parallel x\left(t\right)\parallel \le \parallel x\left(0\right)\parallel e^{-kt},$$

showing exponential convergence under the PL condition.

Compared with classical LaSalle or Matrosov theorems, the proposed composite framework relaxes smoothness and compactness assumptions and provides stronger quantitative outcomes. While traditional approaches conclude only qualitative convergence under uniform continuity and invariance conditions, the present method establishes explicit decay rates of the form $\dot{W}\le -\gamma (N_1+N_2)$ and derives $L^2$ and $O\left(t^{-1/2}\right)$ estimates for $\mathrm{dist}\left(x\right(t),E)$. Hence, it extends the scope of Lyapunov analysis to nonsmooth and noncompact settings while yielding sharper convergence conclusions. It also connects naturally with the Polyak–Łojasiewicz and Kurdyka–Łojasiewicz conditions in variational systems, showing that the strict-decay inequality unifies energy-based and error–bound approaches within a single quantitative framework.

2.6. Pointwise and Set Stability Results

The convergence results established so far show that trajectories approach the critical set E. The next step is to translate this asymptotic behavior into stability properties. Throughout this section, the analysis is carried out on bounded sublevels of the Lyapunov function W, ensuring that all trajectories considered remain bounded. By combining convergence to E with Lyapunov stability of its elements, we obtain pointwise asymptotic stability of the set, semistability of equilibria, and asymptotic stability of an equilibrium for all bounded trajectories (global if W is radially unbounded). These results summarize the qualitative behavior of the system and will later serve as the foundation for the applications.

We recall the notions of pointwise asymptotic stability (PAS) and semistability (SS) used in the statements below.

Definition 1

(Pointwise Asymptotic Stability). A set Z is said to be pointwise asymptotically stable (PAS) for (1) if:

• (${\mathcal{A}}_1$) every $z\in Z$ is Lyapunov stable;
• (${\mathcal{A}}_2$) every solution $x\left(t\right)$ of (1) converges and ${lim}_{t\to \infty }x\left(t\right)\in Z$, i.e., there exists $\delta >0$ such that $\parallel x_0-z\parallel \le \delta$ implies ${lim}_{t\to \infty }x\left(t\right)=z^{\ast }\in Z$.

Definition 2

(Semistability). An equilibrium $x^{\ast }\in E_f$ is said to be semistable (SS) if it satisfies (${\mathcal{A}}_1$) and (${\mathcal{A}}_2$) for $Z=E_f$.

Pointwise asymptotic stability generalizes asymptotic stability from a single equilibrium to a possibly nonisolated set of equilibria. In noncompact or continuous equilibrium sets, classical asymptotic stability is no longer attainable, since nearby equilibria prevent isolation. In such settings, semistability provides the natural notion of convergence: each equilibrium is Lyapunov stable, and trajectories starting near the set converge to one of its elements.

We now state the main stability consequences of the convergence result established above.

Corollary 2

(Pointwise Asymptotic Stability of the Critical Set). Let

$$E:=\{x\in {\mathbb{R}}^n:N_1\left(x\right)=0,N_2\left(x\right)=0\}.$$

If every point of E is Lyapunov stable, then E is pointwise asymptotically stable (in the sense of Definition 1). Equivalently, every trajectory $x(·)$ converges and its limit belongs to E.

Proof. 

By Theorem 3, every trajectory $x(·)$ satisfies $dist\left(x\right(t),E)\to 0$ as $t\to \infty$; hence, the $\omega$–limit set of $x(·)$ is contained in E. This verifies condition (${\mathcal{A}}_2$) of Definition 1. By assumption, each point of E is Lyapunov stable, which is condition (${\mathcal{A}}_1$). Therefore, both conditions hold, and E is pointwise asymptotically stable. □

The next result specializes this conclusion to the case where the critical set coincides with the equilibrium set.

Corollary 3

(Semistability of Equilibria). Suppose that the critical set

$$E:=\{x\in {\mathbb{R}}^n:N_1\left(x\right)=0,N_2\left(x\right)=0\}$$

coincides with the set of equilibria of the system. If every equilibrium in E is Lyapunov stable, then each equilibrium is semistable: every trajectory $x(·)$ converges and its ω–limit set reduces to a single equilibrium in E.

Proof. 

By Theorem 3, every trajectory $x(·)$ satisfies $dist\left(x\right(t),E)\to 0$ as $t\to \infty$; hence, the $\omega$–limit set of $x(·)$ is contained in E. This guarantees convergence to equilibria. Since each equilibrium in E is Lyapunov stable, trajectories starting near an equilibrium cannot drift among different points of E, and thus must converge to a single equilibrium. This establishes semistability. □

Finally, when the critical set reduces to a single equilibrium, the result recovers global asymptotic stability.

Corollary 4

(Asymptotic Stability of an Equilibrium). If the critical set reduces to a singleton

$$E=\left\{x^{\ast }\right\},$$

then $x^{\ast }$ is asymptotically stable for all bounded trajectories. If, in addition, the Lyapunov function W is radially unbounded (i.e., $W\left(x\right)\to \infty$ as $\parallel x\parallel \to \infty$), the result extends to global asymptotic stability.

Proof. 

By Theorem 3, every trajectory $x(·)$ satisfies $dist\left(x\right(t),E)\to 0$ as $t\to \infty$. Since the analysis is carried out on bounded sublevels of W, the trajectories remain bounded. Because E consists of the single equilibrium $x^{\ast }$, this implies $x\left(t\right)\to x^{\ast }$. Hence, $x^{\ast }$ is asymptotically stable for all bounded trajectories; if W is radially unbounded, the stability is global. □

Remark 9

(Set stability versus point stability). Classical Lyapunov theory is typically concerned with the asymptotic stability of an isolated equilibrium point $x^{\ast }$, where stability means that solutions starting near $x^{\ast }$ remain close, and asymptotic stability means they converge to $x^{\ast }$. The framework developed here is more general. The critical set

$$E:=\{x\in {\mathbb{R}}^n:N_1\left(x\right)=0,N_2\left(x\right)=0\}$$

need not be a singleton: it may contain infinitely many equilibria or form a smooth manifold.

This extends the classical notion of asymptotic stability from isolated equilibria to general invariant sets. In such cases, it is natural to ask for the stability of the set E rather than of a single point. The strict-decay inequality ensures that $dist\left(x\right(t),E)\to 0$ as $t\to \infty$, establishing asymptotic stability of the set E. This generalization is particularly relevant for dynamical systems arising in optimization and control, such as the inertial gradient-like system and the Primal–Dual gradient flow studied in the next section.

3. Applications and Refined Results

3.1. Case Study I: Inertial Gradient-like System

In this case, we study the inertial gradient-like system studied in [15]. Compared with their approach, which relies on Opial’s lemma and weak convergence arguments, our framework uses a strict-decay inequality that makes the proof more direct. In addition, this allows us to establish semistability and strong convergence in the convex case. The same inertial gradient-like dynamics have also been recently analyzed using Lyapunov pairs to characterize stability [13], and through refined invariance principles, providing a precise location of the $\omega$-limit set [16].

Let $\Phi :{\mathbb{R}}^n\to \mathbb{R}$ be of class $C^2$, bounded from below, and with Lipschitz continuous Hessian on bounded sets of ${\mathbb{R}}^n$. We consider the second-order dynamical system

$$\ddot{x}\left(t\right)+\alpha \dot{x}\left(t\right)+\beta {\nabla }^2\Phi \left(x\left(t\right)\right)\dot{x}\left(t\right)+\nabla \Phi \left(x\left(t\right)\right)=0,\alpha ,\beta >0.$$

(8)

This system can be equivalently written in the phase space as a first-order Cauchy problem. Introducing the variable $y\left(t\right)=\left(x\right(t),v(t\left)\right)$ with $v\left(t\right)=\dot{x}\left(t\right)$, we obtain

$$\dot{x}=v,\dot{v}=-\alpha v-\nabla \Phi \left(x\right)-\beta {\nabla }^2\Phi \left(x\right)v.$$

The corresponding equilibrium set is $S=\left\{\right(x,v):v=0,\nabla \Phi (x)=0\}$. Moreover, define the function

$$W(x,v)=(\alpha \beta +1)\Phi \left(x\right)+\frac{1}{2}{\parallel v+\beta \nabla \Phi \left(x\right)\parallel }^2.$$

A direct computation (see [15]), yields

$$\dot{W}=-{\alpha \parallel v\parallel }^2-\beta {\parallel \nabla \Phi \left(x\right)\parallel }^2.$$

Introduce the observables

$$N_1(x,v):={\parallel v\parallel }^2,N_2(x,v):={\parallel \nabla \Phi \left(x\right)\parallel }^2,$$

so that

$$\dot{W}=-\alpha N_1-\beta N_2\le 0.$$

Since W is nonincreasing and bounded below, Proposition 1 applies and provides the integral bounds

$${\int }_0^{\infty }{N}_1\left(y\left(s\right)\right)ds<\infty ,{\int }_0^{\infty }{N}_2\left(y\left(s\right)\right)ds<\infty ;$$

hence, $\dot{x}$ and $\nabla \Phi \left(x\right)$ belong to $L^2(0,+\infty )$. Now assume that the trajectory $\left(x\right(t),v(t\left)\right)$ is bounded. Then both $\nabla \Phi$ and ${\nabla }^2\Phi$ are bounded on it. From the dynamics, $\dot{v}$ is bounded, so v is Lipschitz and $N_1$ is uniformly continuous. Moreover,

$${\displaystyle \frac{d}{dt}}\nabla \Phi \left(x\left(t\right)\right)={\nabla }^2\Phi \left(x\left(t\right)\right)v\left(t\right)$$

is bounded; hence, $\nabla \Phi \left(x\right(t\left)\right)$ is Lipschitz and $N_2$ is uniformly continuous. By Proposition 1, the combination of strict decay and uniform continuity yields

$$N_1\left(y\left(t\right)\right)\to 0,N_2\left(y\left(t\right)\right)\to 0,t\to +\infty$$

Therefore,

$$\mathrm{dist}\left(\right(x\left(t\right),v\left(t\right)),S)\to 0.$$

By Corollary 2, this shows the Pointwise Asymptotic Stability (PAS) of the critical set $S=\{y\in {\mathbb{R}}^n\times {\mathbb{R}}^n:N_1\left(y\right)=N_2\left(y\right)=0\}$.

The above analysis shows that bounded trajectories converge to the critical set

$$S=\{(x,v)\in {\mathbb{R}}^n\times {\mathbb{R}}^n:v=0,\nabla \Phi \left(x\right)=0\},$$

establishing pointwise asymptotic stability in the sense of Corollary 2.

To go further, we now specialize in the case where $\Phi$ is convex. In this setting, the condition $\nabla \Phi \left(x\right)=0$ is equivalent to $x\in Argmin\Phi$; thus, the critical set reduces to

$$S=Argmin\Phi \times \left\{0\right\}.$$

Hence, convergence to S ensures that trajectories approach the minimizer set,

$$dist\left(x\right(t),Argmin\Phi )\to 0.$$

While this guarantees convergence to the set of minimizers, it still leaves open the possibility of multiple cluster points within $Argmin\Phi$. To address this and prove convergence to a single minimizer, we refine the Lyapunov construction by introducing a perturbed functional $W_{\epsilon }$. This perturbation anchors the dynamics at a reference minimizer and enables the application of the semistability result provided in Corollary 3.

To strengthen the convergence result in the convex case, we introduce a perturbed Lyapunov functional. Fix a reference point $z\in Argmin\Phi$ and, for small $\epsilon >0$, define

$$W_{\epsilon }(x,v)=W(x,v)+\epsilon \left({\displaystyle \frac{\alpha }{2}}{\parallel x-z\parallel }^2+⟨v+\beta \nabla \Phi \left(x\right),x-z⟩\right).$$

This functional is equivalent to W and remains bounded from below. Differentiating along the trajectories of the system yields

$${\dot{W}}_{\epsilon }=-{\alpha \parallel v\parallel }^2-\beta {\parallel \nabla \Phi \left(x\right)\parallel }^2+\epsilon \left({\parallel v\parallel }^2+\beta ⟨\nabla \Phi \left(x\right),v⟩-⟨\nabla \Phi \left(x\right),x-z⟩\right).$$

Since $\Phi$ is convex and z is a minimizer, the last inner product satisfies $⟨\nabla \Phi \left(x\right),x-z⟩\ge 0$. We may, therefore, drop this term to obtain the upper bound

$${\dot{W}}_{\epsilon }\le -{\alpha \parallel v\parallel }^2-\beta {\parallel \nabla \Phi \left(x\right)\parallel }^2+\epsilon \left({\parallel v\parallel }^2+\beta ⟨\nabla \Phi \left(x\right),v⟩\right).$$

The remaining cross-term can be estimated using Young’s inequality, as follows

$$\beta \left|⟨\nabla \Phi \left(x\right),v⟩\right|\le \frac{1}{2}{\parallel v\parallel }^2+\frac{{\beta }^2}{2}{\parallel \nabla \Phi \left(x\right)\parallel }^2.$$

For $0<\epsilon <min\left\{\frac{2}{3}\alpha ,\frac{2}{\beta }\right\}$ and substituting the previous estimate leads to

$${\dot{W}}_{\epsilon }\le -\left(\alpha -\frac{3}{2}\epsilon \right){\parallel v\parallel }^2-\left(\beta -\frac{{\beta }^2}{2}\epsilon \right){\parallel \nabla \Phi \left(x\right)\parallel }^2.$$

Therefore, there exists

$$c_{\epsilon }:=min\left\{\alpha -\frac{3}{2}\epsilon ,\beta -\frac{{\beta }^2}{2}\epsilon \right\}>0,$$

so that

$${\dot{W}}_{\epsilon }\le -c_{\epsilon }\left({\parallel v\parallel }^2+{\parallel \nabla \Phi \left(x\right)\parallel }^2\right)=-c_{\epsilon }(N_1+N_2).$$

This inequality shows that $W_{\epsilon }$ is nonincreasing and convergent. In particular, $\dot{x}$ and $\nabla \Phi \left(x\right)$ both belong to $L^2(0,\infty )$. Since the trajectory is bounded, the functions $N_1$ and $N_2$ are uniformly continuous along the trajectory, and Proposition 1 guarantees that

$$v\left(t\right)\to 0\mathrm{and}\nabla \Phi \left(x\right(t\left)\right)\to 0\mathrm{as}t\to \infty .$$

It remains to identify the limiting point. By construction, the difference between $W_{\epsilon }$ and W is

$$W_{\epsilon }\left(t\right)-W\left(t\right)=\epsilon \left({\displaystyle \frac{\alpha }{2}}{\parallel x\left(t\right)-z\parallel }^2+⟨v\left(t\right)+\beta \nabla \Phi \left(x\left(t\right)\right),x\left(t\right)-z⟩\right).$$

Both $W_{\epsilon }$ and W converge to finite limits; thus, their difference does as well. The inner product term vanishes asymptotically because $v\left(t\right)\to 0$ and $\nabla \Phi \left(x\right(t\left)\right)\to 0$. Consequently, the limit of $\parallel x\left(t\right)-z\parallel$ exists for every minimizer $z\in Argmin\Phi$.

Since $x\left(t\right)$ is bounded, it admits cluster points. Every cluster point lies in $Argmin\Phi$, but the existence of ${lim}_{t\to \infty }\parallel x\left(t\right)-z\parallel$ for all z forces the cluster set to be a singleton. Thus, the trajectory converges strongly to one minimizer,

$$x\left(t\right)\to x^{\ast }\in Argmin\Phi .$$

In conclusion, the perturbed Lyapunov functional $W_{\epsilon }$ establishes convergence of every bounded trajectory to a single minimizer. This corresponds to Semistability (SS) of the equilibria in the sense of Corollary 3: each equilibrium in E is Lyapunov stable, and every trajectory converges to one equilibrium. If the minimizer set is reduced to a singleton, then semistability and pointwise asymptotic stability coincide, and we recover classical asymptotic stability of the unique equilibrium (see Corollary 4).

3.2. Case Study II: Primal–Dual Gradient Flow

We now consider the Primal–Dual gradient flow, a continuous-time model for equality-constrained convex optimization problems. It is used in communication networks (resource allocation, congestion control), power systems, and decentralized control, and in analyses of first-order saddle-point methods for machine learning and inverse problems. Many works study this flow (see, for example, [17,18,19,20])

Here we treat the baseline model within our strict-decay framework and obtain asymptotic stability of the Karush–Kuhn–Tucker (KKT) set, with a semistability refinement.

Consider the equality-constrained convex optimization problem

$$\underset{x\in {\mathbb{R}}^n}{min}\Phi \left(x\right)\mathrm{subject}\mathrm{to}Ax=b,$$

where $\Phi :{\mathbb{R}}^n\to \mathbb{R}$ is convex and continuously differentiable with locally Lipschitz gradient, $A\in {\mathbb{R}}^{m\times n}$, and $b\in {\mathbb{R}}^m$.

Throughout this subsection, we assume that the constraint is feasible, i.e., $Ax=b$ admits at least one solution. If $Ax=b$ is infeasible, the KKT set is empty and the standard Primal–Dual flow may produce divergent multipliers; thus, the stability results below do not apply.

The Lagrangian is

$$L(x,\lambda )=\Phi \left(x\right)+{\lambda }^{\top }(Ax-b),$$

with multiplier $\lambda \in {\mathbb{R}}^m$. The Karush–Kuhn–Tucker conditions are

$${\nabla }_{x}L(x,\lambda )=\nabla \Phi \left(x\right)+A^{\top }\lambda =0,{\nabla }_{\lambda }L(x,\lambda )=Ax-b=0.$$

Any pair $(x^{\ast },{\lambda }^{\ast })$ satisfying them is a saddle point of L and a KKT equilibrium.

A natural continuous-time approach is the Primal–Dual (Arrow–Hurwicz–Uzawa) gradient flow:

$$\dot{x}=-{\nabla }_{x}L(x,\lambda ),\dot{\lambda }={\nabla }_{\lambda }L(x,\lambda ),$$

i.e.,

$$\dot{x}=-\nabla \Phi \left(x\right)-A^{\top }\lambda ,\dot{\lambda }=Ax-b.$$

This system, known as the Arrow–Hurwicz–Uzawa flow (see [21]), represents a continuous-time analog of the classical Primal–Dual algorithm. The variable x seeks to minimize the Lagrangian, while $\lambda$ enforces the constraint by driving the residual $Ax-b$ to zero.

The equilibrium (KKT) set is

$$S=\{(x,\lambda ):\nabla \Phi \left(x\right)+A^{\top }\lambda =0,Ax=b\}.$$

For stability, use the Lyapunov functional

$$W(x,\lambda )=\Phi \left(x\right)-\Phi \left(x^{\ast }\right)+\frac{1}{2}{\parallel Ax-b\parallel }^2+\frac{1}{2}{\parallel \lambda -{\lambda }^{\ast }\parallel }^2,$$

with $(x^{\ast },{\lambda }^{\ast })\in S$. Along trajectories,

$$\dot{W}={-\parallel Ax-b\parallel }^2-{\parallel \nabla \Phi \left(x\right)+A^{\top }\lambda \parallel }^2.$$

Define

$$N_1(x,\lambda )={\parallel Ax-b\parallel }^2,N_2(x,\lambda )={\parallel \nabla \Phi \left(x\right)+A^{\top }\lambda \parallel }^2,$$

Then

$$\dot{W}=-N_1-N_2.$$

This is a strict-decay identity in our framework. Since W is nonincreasing and bounded below, we have

$${\int }_0^{\infty }{N}_1(x\left(t\right),\lambda \left(t\right))dt<\infty ,{\int }_0^{\infty }{N}_2(x\left(t\right),\lambda \left(t\right))dt<\infty .$$

Since W contains the quadratic term $\frac{1}{2}{\parallel \lambda -{\lambda }^{\ast }\parallel }^2$, any bounded sublevel of W ensures bounded multipliers $\lambda \left(t\right)$. We keep this assumption implicit and focus on the feasible case. If $\left(x\right(t),\lambda (t\left)\right)$ is bounded, local Lipschitzness of the vector field makes $N_1$ and $N_2$ uniformly continuous along the trajectory. By the integral-vanishing principle provided in Proposition 1,

$$N_1\left(t\right)\to 0,N_2\left(t\right)\to 0;$$

thus,

$$Ax\left(t\right)\to b,\nabla \Phi \left(x\left(t\right)\right)+A^{\top }\lambda \left(t\right)\to 0.$$

Therefore,

$$dist\left(\right(x\left(t\right),\lambda \left(t\right)),S)\to 0,$$

establishing asymptotic stability of S. With a unique constrained minimizer, the trajectory converges to that KKT point; otherwise, it converges to the equilibrium set.

If the constrained minimizer is unique, then $(x\left(t\right),\lambda \left(t\right))\to (x^{\ast },{\lambda }^{\ast })$. A sufficient condition is that $\Phi$ is strictly convex on the affine subspace $\{x:Ax=b\}$ and A has full row rank, which also yields uniqueness of ${\lambda }^{\ast }$. If several constrained minimizers exist, trajectories converge to the entire KKT set S, showing semistability.

To capture semistability, perturb W by a skew term that couples primal and dual residuals. For $\epsilon >0$, set

$$W_{\epsilon }(x,\lambda ):=W(x,\lambda )+\epsilon \left(⟨x-x^{\ast },\nabla \Phi \left(x\right)+A^{\top }\lambda ⟩-⟨\lambda -{\lambda }^{\ast },Ax-b⟩\right).$$

On bounded sublevels of W, the gradient $\nabla \Phi$ is L-Lipschitz; thus, $W_{\epsilon }$ and W are equivalent for small $\epsilon$.

To estimate ${\dot{W}}_{\epsilon }$, note that since $\nabla \Phi$ is L-Lipschitz and $x(·)$ is absolutely continuous,

$$\parallel \frac{d}{dt}\nabla \Phi \left(x\left(t\right)\right)\parallel =\underset{h\to 0}{lim}{\displaystyle \frac{\parallel \nabla \Phi \left(x\right(t+h\left)\right)-\nabla \Phi \left(x\right(t\left)\right)\parallel }{\left|h\right|}}\le L\parallel \dot{x}\left(t\right)\parallel =L\parallel \nabla \Phi \left(x\right)+A^{\top }\lambda \parallel \mathrm{for}\mathrm{a}.\mathrm{e}.t.$$

Using the system dynamics and this inequality, we have

$$\begin{array}{cc}\hfill {\displaystyle \frac{d}{dt}}⟨x-x^{\ast },\nabla \Phi \left(x\right)+A^{\top }\lambda ⟩& =⟨\dot{x},\nabla \Phi \left(x\right)+A^{\top }\lambda ⟩+⟨x-x^{\ast },\frac{d}{dt}\nabla \Phi \left(x\right)+A^{\top }\dot{\lambda }⟩\hfill \\ & =-\parallel \nabla \Phi \left(x\right)+A^{\top }{\lambda \parallel }^2+⟨x-x^{\ast },A^{\top }(Ax-b)⟩+⟨x-x^{\ast },\frac{d}{dt}\nabla \Phi \left(x\right)⟩,\hfill \\ \hfill {\displaystyle \frac{d}{dt}}⟨\lambda -{\lambda }^{\ast },Ax-b⟩& =⟨\dot{\lambda },Ax-b⟩+⟨\lambda -{\lambda }^{\ast },A\dot{x}⟩\hfill \\ & ={\parallel Ax-b\parallel }^2-⟨\lambda -{\lambda }^{\ast },A(\nabla \Phi \left(x\right)+A^{\top }\lambda )⟩.\hfill \end{array}$$

Substituting these expressions into ${\dot{W}}_{\epsilon }$ yields

$$\begin{array}{cc}\hfill {\dot{W}}_{\epsilon }& =\dot{W}+\epsilon \left(\frac{d}{dt}⟨x-x^{\ast },·⟩-\frac{d}{dt}⟨\lambda -{\lambda }^{\ast },·⟩\right)\hfill \\ & ={-(1+\epsilon )\parallel Ax-b\parallel }^2-(1+\epsilon ){\parallel \nabla \Phi \left(x\right)+A^{\top }\lambda \parallel }^2\hfill \\ & +\epsilon \left(⟨x-x^{\ast },A^{\top }(Ax-b)⟩-⟨\lambda -{\lambda }^{\ast },A(\nabla \Phi \left(x\right)+A^{\top }\lambda )⟩+⟨x-x^{\ast },\frac{d}{dt}\nabla \Phi \left(x\right)⟩\right).\hfill \end{array}$$

On a bounded sublevel of W, we estimate the cross-terms using the Cauchy–Schwarz and Young inequalities (${\displaystyle ab\le 2\eta a^2+\frac{1}{8\eta }{b}^2}$ with $\eta >0$):

$$\begin{array}{cc}\hfill |⟨x-x^{\ast },A^{\top }(Ax-b)⟩|& \le \parallel A\parallel \parallel x-x^{\ast }\parallel \parallel Ax-b\parallel \le 2{\eta }_1{\parallel Ax-b\parallel }^2+{\displaystyle \frac{{\parallel A\parallel }^2}{8{\eta }_1}}{\parallel x-x^{\ast }\parallel }^2,\hfill \\ \hfill |⟨\lambda -{\lambda }^{\ast },A(\nabla \Phi \left(x\right)+A^{\top }\lambda )⟩|& \le \parallel A\parallel \parallel \lambda -{\lambda }^{\ast }\parallel \parallel \nabla \Phi \left(x\right)+A^{\top }\lambda \parallel \hfill \\ \hfill & \le 2{\eta }_2\parallel \nabla \Phi \left(x\right)+A^{\top }{\lambda \parallel }^2+{\displaystyle \frac{{\parallel A\parallel }^2}{8{\eta }_2}}{\parallel \lambda -{\lambda }^{\ast }\parallel }^2,\hfill \\ \hfill |⟨x-x^{\ast },\frac{d}{dt}\nabla \Phi \left(x\right)⟩|& \le \parallel x-x^{\ast }\parallel \parallel \frac{d}{dt}\nabla \Phi \left(x\right)\parallel \le L\parallel x-x^{\ast }\parallel \parallel \nabla \Phi \left(x\right)+A^{\top }\lambda \parallel \hfill \\ \hfill & \le 2{\eta }_3\parallel \nabla \Phi \left(x\right)+A^{\top }{\lambda \parallel }^2+{\displaystyle \frac{L^2}{8{\eta }_3}}{\parallel x-x^{\ast }\parallel }^2.\hfill \end{array}$$

Inserting these bounds into the previous inequality yields

$$\begin{array}{cc}\hfill {\dot{W}}_{\epsilon }& \le {-(1+\epsilon )\parallel Ax-b\parallel }^2-(1+\epsilon ){\parallel \nabla \Phi \left(x\right)+A^{\top }\lambda \parallel }^2\hfill \\ & +\epsilon \left(2{\eta }_1{\parallel Ax-b\parallel }^2+2({\eta }_2+{\eta }_3){\parallel \nabla \Phi \left(x\right)+A^{\top }\lambda \parallel }^2\right)\hfill \\ & +\epsilon \left({\displaystyle \frac{{\parallel A\parallel }^2}{8{\eta }_1}}\parallel x-x^{\ast }{\parallel }^2+{\displaystyle \frac{{\parallel A\parallel }^2}{8{\eta }_2}}\parallel \lambda -{\lambda }^{\ast }{\parallel }^2+{\displaystyle \frac{L^2}{8{\eta }_3}}{\parallel x-x^{\ast }\parallel }^2\right).\hfill \end{array}$$

Collecting coefficients gives

$${\dot{W}}_{\epsilon }\le -\left(1+\epsilon -2\epsilon {\eta }_1\right){\parallel Ax-b\parallel }^2-\left(1+\epsilon -2\epsilon ({\eta }_2+{\eta }_3)\right){\parallel \nabla \Phi \left(x\right)+A^{\top }\lambda \parallel }^2+\epsilon R(x,\lambda ),$$

where

$$R(x,\lambda )={\displaystyle \frac{{\parallel A\parallel }^2}{8{\eta }_1}}\parallel x-x^{\ast }{\parallel }^2+{\displaystyle \frac{{\parallel A\parallel }^2}{8{\eta }_2}}\parallel \lambda -{\lambda }^{\ast }{\parallel }^2+{\displaystyle \frac{L^2}{8{\eta }_3}}{\parallel x-x^{\ast }\parallel }^2.$$

To control R by W, note that since $x^{\ast }$ minimizes $\Phi$, we have $\Phi \left(x\right)-\Phi \left(x^{\ast }\right)\ge 0$, and therefore $W(x,\lambda )\ge 0$. Moreover,

$$\parallel \lambda -{\lambda }^{\ast }{\parallel }^2\le {2W(x,\lambda ),\parallel Ax-b\parallel }^2=\parallel A(x-x^{\ast }){\parallel }^2\le {\parallel A\parallel }^2{\parallel x-x^{\ast }\parallel }^2.$$

Working on a bounded sublevel $\{W\le W_0\}$, which is compact since the trajectory is bounded, there exist ${\alpha }^{\prime }>0$ and $r>0$ such that

$$\parallel x-x^{\ast }{\parallel }^2\le {\displaystyle \frac{1}{{\alpha }^{\prime }}}W(x,\lambda )\mathrm{for}\parallel (x,\lambda )-(x^{\ast },{\lambda }^{\ast })\parallel \le r.$$

Outside this neighborhood, continuity on the compact set implies

$$\delta :=\underset{K}{min}W>0,M:=\underset{\{W\le W_0\}}{max}\parallel x-x^{\ast }\parallel <\infty ,$$

so that $\parallel x-x^{\ast }{\parallel }^2\le (M^2/\delta )W(x,\lambda )$ for all $(x,\lambda )\in K$. Setting

$$C_x=max\left\{{\displaystyle \frac{1}{{\alpha }^{\prime }}},{\displaystyle \frac{M^2}{\delta }}\right\},$$

we have $\parallel x-x^{\ast }{\parallel }^2\le C_{x}W(x,\lambda )$ on $\{W\le W_0\}$, and thus

$$R(x,\lambda )\le \left({\displaystyle \frac{{\parallel A\parallel }^2}{8{\eta }_1}}+{\displaystyle \frac{L^2}{8{\eta }_3}}\right)C_{x}W(x,\lambda )+{\displaystyle \frac{{\parallel A\parallel }^2}{8{\eta }_2}}·2W(x,\lambda )=c_{0}W(x,\lambda ),$$

where

$$c_0=\left({\displaystyle \frac{{\parallel A\parallel }^2}{8{\eta }_1}}+{\displaystyle \frac{L^2}{8{\eta }_3}}\right)C_x+{\displaystyle \frac{{\parallel A\parallel }^2}{4{\eta }_2}}.$$

Hence,

$${\dot{W}}_{\epsilon }\le -a_1{\parallel Ax-b\parallel }^2-a_2{\parallel \nabla \Phi \left(x\right)+A^{\top }\lambda \parallel }^2+\epsilon c_{0}W,\left\{\begin{array}{c}{a}_1=1+\epsilon -2\epsilon {\eta }_1,\hfill \\ a_2=1+\epsilon -2\epsilon ({\eta }_2+{\eta }_3).\hfill \end{array}\right.$$

On the same compact set, there exists $\kappa >0$ such that

$$W(x,\lambda )\le \kappa \left({\parallel Ax-b\parallel }^2+{\parallel \nabla \Phi \left(x\right)+A^{\top }\lambda \parallel }^2\right),$$

since both sides are continuous and vanish only at $(x^{\ast },{\lambda }^{\ast })$. Consequently,

$$\epsilon c_{0}W\le \epsilon c_0\kappa \left({\parallel Ax-b\parallel }^2+{\parallel \nabla \Phi \left(x\right)+A^{\top }\lambda \parallel }^2\right).$$

Choosing $\epsilon >0$ sufficiently small so that $\epsilon c_0\kappa \le \frac{1}{2}min\{a_1,a_2\}$, we obtain

$${\dot{W}}_{\epsilon }\le -(a_1-\epsilon c_0\kappa ){\parallel Ax-b\parallel }^2-(a_2-\epsilon c_0\kappa ){\parallel \nabla \Phi \left(x\right)+A^{\top }\lambda \parallel }^2.$$

Both coefficients are strictly positive, setting

$$c_1=a_1-\epsilon c_0\kappa >0,c_2=a_2-\epsilon c_0\kappa >0,$$

we finally obtain

$${\dot{W}}_{\epsilon }\le -c_1{\parallel Ax-b\parallel }^2-c_2{\parallel \nabla \Phi \left(x\right)+A^{\top }\lambda \parallel }^2.$$

This shows that ${\dot{W}}_{\epsilon }$ is strictly negative outside equilibrium along bounded trajectories.

To conclude, the perturbed Lyapunov function $W_{\epsilon }$ is positive definite with respect to the KKT set S and strictly decreases along trajectories outside S. Therefore, every KKT equilibrium is Lyapunov stable, and

$$\parallel Ax\left(t\right)-b\parallel \to 0,\parallel \nabla \Phi \left(x\left(t\right)\right)+A^{\top }\lambda \left(t\right)\parallel \to 0,$$

whence

$$dist\left(\right(x\left(t\right),\lambda \left(t\right)),S)\to 0.$$

Thus, S is semistable. If the constrained minimizer of $\Phi$ subject to $Ax=b$ is unique, the trajectory converges to that single KKT point.

Illustrative Example

To illustrate the result, consider the quadratic optimization problem

$$\begin{array}{cc}\hfill \mathrm{minimize}& \Phi \left(x\right)=\frac{1}{2}{\parallel x\parallel }^2,\hfill \\ \hfill \mathrm{subject}\mathrm{to}& x_1+x_2=1.\hfill \end{array}$$

Here, $A=\left[11\right]$ and $b=1$. The Karush–Kuhn–Tucker (KKT) conditions are

$$x+A^{\top }\lambda =0,Ax=b.$$

Solving these equations yields the unique KKT point

$$x^{\ast }=(0.5,0.5),{\lambda }^{\ast }=-0.5.$$

The corresponding Primal–Dual flow is

$$\dot{x}=-x-A^{\top }\lambda ,\dot{\lambda }=Ax-b.$$

For this system, the Lyapunov function

$$W(x,\lambda )=\Phi \left(x\right)-\Phi \left(x^{\ast }\right)+\frac{1}{2}{\parallel Ax-b\parallel }^2+\frac{1}{2}{\parallel \lambda -{\lambda }^{\ast }\parallel }^2$$

satisfies

$$\dot{W}={-\parallel Ax-b\parallel }^2-{\parallel x+A^{\top }\lambda \parallel }^2\le 0.$$

Hence, both residuals

$$r_p\left(t\right)=Ax\left(t\right)-b,r_d\left(t\right)=x\left(t\right)+A^{\top }\lambda \left(t\right),$$

tend to zero as $t\to \infty$. Therefore, the trajectory $\left(x\right(t),\lambda (t\left)\right)$ converges to the KKT point $(x^{\ast },{\lambda }^{\ast })$, confirming the strict-decay property and asymptotic stability predicted by the theory.

4. Conclusions

The paper developed a composite Lyapunov framework that establishes strict decay through a pair of differential inequalities. This construction provides a unified and generalized structure that encompasses both the classical Lyapunov method—based on single-function decay—and the Matrosov auxiliary–function framework, where multiple functions interact to ensure convergence. By merging these two perspectives into a single principle of composite decay inequalities, the approach delivers a direct and constructive path from decay estimates to integral bounds, convergence to the critical set, and stability of equilibria and invariant sets, all without relying on compactness or invariance assumptions. Applications to inertial gradient-like systems and Primal–Dual gradient flows illustrate how this unified framework captures semistability and convergence in optimization-driven dynamics while offering a transparent analytical tool for broader classes of nonlinear systems.