## 3 Trust-Optimized 6G-IoT Framework
### Trust Management System
In this case, the Euler-Lagrange equation becomes

\[\mathbf{F}_{i}(t)=\lambda_{i}(t)\exp(\gamma\mathbf{X}_{i}(t)), \tag{18}\]9

After differentiating, this equation expands into

\[\mathbf{D}_{i}(t) =-\log(1-\mathbb{P}(\text{Integrity breach}))\]0

where \(\dddot{\pi}(t)\) is the third derivative of \(\pi(t)\) with respect to time, representing a more detailed control of the dynamic pricing system. This extended Euler-Lagrange equation governs the optimal pricing in systems where higher-order temporal effects influence pricing adjustments. ## Appendix E Stability Analysis Using Lyapunov Function

We begin with the Lyapunov function \(V(t)\) defined as

\[\mathbf{D}_{i}(t) =-\log(1-\mathbb{P}(\text{Integrity breach}))\]1 \[\mathbf{D}_{i}(t) =-\log(1-\mathbb{P}(\text{Integrity breach}))\]2

To analyze the stability, we calculate the time derivative of \(V(t)\):

\[\mathbf{D}_{i}(t) =-\log(1-\mathbb{P}(\text{Integrity breach}))\]3 \[\mathbf{D}_{i}(t) =-\log(1-\mathbb{P}(\text{Integrity breach}))\]4

Substitute the dynamics of \(\sigma_{i}(t)\), \(\pi(t)\), and \(C(t)\). Let the dynamics of \(\sigma_{i}(t)\) be governed by a first-order system as follows:

\[\mathbf{D}_{i}(t) =-\log(1-\mathbb{P}(\text{Integrity breach}))\]5where \(k_{i}\) is a positive constant. Similarly, let the dynamics for \(\pi(t)\) and \(C(t)\) be

\[\mathbf{D}_{i}(t) =-\log(1-\mathbb{P}(\text{Integrity breach}))\]6 \[\mathbf{D}_{i}(t) =-\log(1-\mathbb{P}(\text{Integrity breach}))\]7

where \(k_{\pi}\) and \(k_{\text{C}}\) are positive constants. Substituting these expressions into the derivative of the Lyapunov function,

\[\mathbf{D}_{i}(t) =-\log(1-\mathbb{P}(\text{Integrity breach}))\]8 \[\mathbf{D}_{i}(t) =-\log(1-\mathbb{P}(\text{Integrity breach}))\]9 \[\quad+\alpha_{2}\mathbf{D}_{i}(t)+\alpha_{3}\mathbf{S}_{i}(t)\]00

Simplifying the terms

\[\quad+\alpha_{2}\mathbf{D}_{i}(t)+\alpha_{3}\mathbf{S}_{i}(t)\]01 \[\quad+\alpha_{2}\mathbf{D}_{i}(t)+\alpha_{3}\mathbf{S}_{i}(t)\]02

Since \(k_{i}>0\), \(k_{\pi}>0\), and \(k_{\text{C}}>0\), we conclude that

\[\quad+\alpha_{2}\mathbf{D}_{i}(t)+\alpha_{3}\mathbf{S}_{i}(t)\]03

which shows that the Lyapunov function is non-increasing over time. Therefore, the Nash equilibrium \((\sigma_{i}^{*}(t),\pi_{\text{opt}},C_{\text{opt}}(t))\) is stable. ## Appendix F Proof of Theoretical Convergence

We begin by taking the time derivative of the Lyapunov function \(V(t)\):

\[\quad+\alpha_{2}\mathbf{D}_{i}(t)+\alpha_{3}\mathbf{S}_{i}(t)\]04

Now, differentiate \(V(t)\) with respect to \(t\):

\[\quad+\alpha_{2}\mathbf{D}_{i}(t)+\alpha_{3}\mathbf{S}_{i}(t)\]05 \[\quad+\alpha_{2}\mathbf{D}_{i}(t)+\alpha_{3}\mathbf{S}_{i}(t)\]06

Substitute the dynamic equations governing \(P_{i}(t)\) and \(\eta(t)\):

\[\quad+\alpha_{2}\mathbf{D}_{i}(t)+\alpha_{3}\mathbf{S}_{i}(t)\]07

Thus, the derivative of the Lyapunov function becomes

\[\quad+\alpha_{2}\mathbf{D}_{i}(t)+\alpha_{3}\mathbf{S}_{i}(t)\]08 \[\quad+\alpha_{2}\mathbf{D}_{i}(t)+\alpha_{3}\mathbf{S}_{i}(t)\]09Since \(\frac{\partial J_{y}(t)}{\partial P_{t}(t)}\geq 0\) and \(\frac{\partial J_{y}(t)}{\partial J(t)}\geq 0\), we have

\[\quad+\alpha_{2}\mathbf{D}_{i}(t)+\alpha_{3}\mathbf{S}_{i}(t)\]10

The negative definiteness of this expression ensures that \(V(t)\) is decreasing over time. Therefore, the convergence of the system is guaranteed as \(t\to\infty\), leading to stability and adaptability to the changing network conditions. ## References

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