## 3 Trust-Optimized 6G-IoT Framework
### Trust Management System
The Euler-Lagrange equation (Appendix D) for this problem is

\[\quad+\alpha_{2}\mathbf{D}_{i}(t)+\alpha_{3}\mathbf{S}_{i}(t)\]7

where \(\mathcal{L}\) is the Lagrangian of the system:

\[\quad+\alpha_{2}\mathbf{D}_{i}(t)+\alpha_{3}\mathbf{S}_{i}(t)\]8

#### 3.3.2 Cost-Efficiency Algorithms

The cost-efficiency of the network is optimized by minimizing the total cost \(C(t)\), which includes both operational \(C_{\text{op}}(t)\) and capital expenditures \(C_{\text{cap}}(t)\):

\[\quad+\alpha_{2}\mathbf{D}_{i}(t)+\alpha_{3}\mathbf{S}_{i}(t)\]9

Operational costs are modeled as

\[\quad+\alpha_{4}\mathbf{C}_{i}(t)+\alpha_{5}\mathbf{I}_{i}(t)+ \alpha_{6}\mathbf{P}_{i}(t), \tag{13}\]0

where \(\kappa\) is a baseline cost factor and \(\mu\) is a scaling factor. Capital expenditures are modeled as

\[\quad+\alpha_{4}\mathbf{C}_{i}(t)+\alpha_{5}\mathbf{I}_{i}(t)+ \alpha_{6}\mathbf{P}_{i}(t), \tag{13}\]1

where \(\delta(t)\) is the depreciation rate and \(\
u\) penalizes fluctuations in investment \(I(t)\). The total cost over time \(T\) is

\[\quad+\alpha_{4}\mathbf{C}_{i}(t)+\alpha_{5}\mathbf{I}_{i}(t)+ \alpha_{6}\mathbf{P}_{i}(t), \tag{13}\]2

The optimization problem is to minimize \(C_{\text{total}}\) subject to constraints on \(S(t)\), \(\gamma(t)\), and \(I(t)\). The first-order conditions are derived using the calculus of variations:

\[\quad+\alpha_{4}\mathbf{C}_{i}(t)+\alpha_{5}\mathbf{I}_{i}(t)+ \alpha_{6}\mathbf{P}_{i}(t), \tag{13}\]3where \(\mathcal{L}\) now represents the Lagrangian for cost-efficiency. The optimization process for cost-efficiency in the proposed framework is outlined in Algorithm 2. ```
Input: Operational parameters \(\kappa\), \(\mu\), Depreciation rate \(\delta(t)\), Investment fluctuation penalty \(\
u\), Time horizon \(T\) Output: Optimized operational strategy \(S(t)\), \(\gamma(t)\), and \(I(t)\) minimizing \(C_{\text{total}}\)
1 Initialization: Set initial conditions for \(S(t)\), \(\gamma(t)\), and \(I(t)\); Initialize cost parameters \(\kappa\), \(\mu\), \(\delta(t)\), \(\
u\);
2 Step 1: Define Total Cost Function
3 Compute total cost \(C(t)\) using (27), with operational cost \(C_{\text{op}}(t)\) defined by (28) and capital expenditures \(C_{\text{cap}}(t)\) defined by (29). 4 Step 2: Derive Euler-Lagrange Conditions
5 Formulate the total cost over time \(T\) as \(C_{\text{total}}\) using (30). Derive the Euler-Lagrange equations using (31) to determine the optimal conditions for \(S(t)\), \(\gamma(t)\), and \(I(t)\). 6 Step 3: Optimize Total Cost
7 Solve the Euler-Lagrange equations iteratively:
8 Update \(S(t)\), \(\gamma(t)\), and \(I(t)\) using numerical methods (e.g., gradient descent). 9 Enforce constraints on \(S(t)\), \(\gamma(t)\), and \(I(t)\) to ensure feasibility. 10 Adjust cost parameters \(\kappa\), \(\mu\), \(\delta(t)\), and \(\
u\) iteratively to minimize \(C_{\text{total}}\). 11 Evaluate convergence criteria to confirm optimality. ```

**Algorithm 2**Cost-efficient optimization algorithm in TEO-IoT framework. #### 3.3.3 Incentive Mechanisms

The designed mechanism of incentives is a way to ensure that at every step the selected strategy by devices \(\sigma_{i}(t)\) maximizes the utilities \(U_{i}(t)\) while being aligned well with the overall network goals. The utility function for each device is

\[\quad+\alpha_{4}\mathbf{C}_{i}(t)+\alpha_{5}\mathbf{I}_{i}(t)+ \alpha_{6}\mathbf{P}_{i}(t), \tag{13}\]4

The Nash equilibrium is found by solving

\[\quad+\alpha_{4}\mathbf{C}_{i}(t)+\alpha_{5}\mathbf{I}_{i}(t)+ \alpha_{6}\mathbf{P}_{i}(t), \tag{13}\]5

The stability of the Nash equilibrium is analyzed using the Lyapunov function:

\[\quad+\alpha_{4}\mathbf{C}_{i}(t)+\alpha_{5}\mathbf{I}_{i}(t)+ \alpha_{6}\mathbf{P}_{i}(t), \tag{13}\]6 \[\quad+\alpha_{4}\mathbf{C}_{i}(t)+\alpha_{5}\mathbf{I}_{i}(t)+ \alpha_{6}\mathbf{P}_{i}(t), \tag{13}\]7

The time derivative of



If \(\frac{dV(t)}{dt}<0\), then the system is stable, it will converge to the optimum economic state. This will form a certain concrete approach toward optimized economics in 6G-enabled IoT networks.