## 2 Problem Formulation and Modeling
### Time-Invariant Parameters in Optimal Control  
When choosing a class of control laws as a reference, we give preference to optimal control. Optimal control is common and representative in nature and practical engineering applications. In practical engineering in the aerospace field, spacecraft orbit transfer relies on optimal control to determine the optimal thrust direction and magnitude, minimize fuel consumption and mission time, and enable spacecraft attitude control. Aircraft flight control uses optimal control to optimize flight trajectories, fuel consumption, and time. We use the following assumptions to standardize the dead reckoning objects discussed next. **Assumption 1**. : _Moving entities possess the capabilities of route planning and motion control._  
**Assumption 2**. : _The moving entity can independently establish the desired target state \(x(t_{f})\) on its own. However, it will not take the initiative to notify the observing side._  
**Assumption 3**. : _The entity (autonomous mobile robots or drones) adopts optimal control in the motion planning of the flight route (continuous state control). Let the entity state be \(x\in\mathbb{R}^{n}\), the control input be \(u\in\mathbb{R}^{p}\), and the corresponding state equation be \(\dot{X}=f(x,u,t)\). The performance index of the optimal state control is described as follows:_  
\[J=\underbrace{\Theta\Big{[}x\Big{(}t_{f}\Big{)},t_{f}\Big{]}}_{\text{End state error}}+\underbrace{\int_{t_{0}}^{t_{f}}F(x,u,t)dt}_{\text{Energy consumption}} \tag{3}\]  
_where \(\Theta(\cdot)\) and \(F(\cdot)\) are scalar functions, representing the positive definite expressions of the state error and the input, respectively._  
To facilitate the analysis and discussion, it is better to simplify the problem to the optimal guidance control in a two-dimensional plane. In Figure 3, the position at time \(t\), namely \(x(t),y(t)\), is located at the lower left corner. The angle between the moving velocity \(V\) and the horizontal axis of the coordinate system is \(\theta_{t}\). The azimuth angle \(q_{t}\) is formed by the desired waypoint \(x(t_{f}),y(t_{f})\) relative to the current position of the moving entity, and \(\theta_{f}\) represents the velocity direction constraint when reaching the desired position in the constraints. \[\left\{ \begin{array}{l}\dot{r}=-V\cdot cos(\theta-q(t))\\ r\dot{q}=-V\cdot sin(\theta-q(t))\end{array} \right. \tag{4}\]  
where  
\[q(t)=tan^{-1}\frac{y(t_{f})-y(t)}{x(t_{f})-x(t)}\]\[r=\sqrt{(y(t_{f})-y(t))^{2}+(x(t_{f})-x(t))^{2}}\]  
Let \(x_{1}=\theta_{f}-q\), \(x_{2}=\dot{q}\); it can be obtained that  
\[\left\{ \begin{array}{l}\dot{x}_{1}=-x_{2}\\ \dot{x}_{2}=(-2t/r)x_{2}+(\dot{r}/r)\dot{\theta}\end{array} \right. \tag{5}\]  
\[\text{Set }X=\left[ \begin{array}{cc}x_{1}&x_{2}\end{array} \right]^{T}\!,A= \left[ \begin{array}{cc}0&-1\\ 0&-2\dot{r}/r\end{array} \right]\!,B=\left[ \begin{array}{cc}0&\dot{r}/r\end{array}  \right]^{T}\!,\text{and }U(t)=\dot{\theta}\text{; thus, the state-space expression of the control system can be obtained,  
\[\left\{ \begin{array}{l}\dot{X}=AX+BU\\ X(t_{f})=0\end{array} \right. \tag{6}\]  
The scalar functions of \(\theta(\cdot)\) and \(F(\cdot)\) in Equation (3) are designed as matrices of parameters \(Q\) and \(R\) of quadratic functional functions and the indicator becomes  
\[J=X^{T}(t_{f})QX(t_{f})+\int_{t_{0}}^{t_{f}}U^{T}(t)RU(t)dt \tag{7}\]  
To solve the optimal control problem of a second-order system using Pontryagin's Minimum Principle, the control law should take the following form,  
\[U(t)=-R^{-1}B^{T}PX \tag{8}\]  
The optimal guidance law corresponding to the solution of the Riccati equation based on the parameter matrix \(Q\) and \(R\) should take the following form in Equation (9):  
\[\dot{\theta}=\underbrace{\mu\cdot\dot{q}}_{\text{Move to }x_{f},y_{f}}+ \underbrace{\lambda(\theta_{f}-q)\cdot\frac{\dot{r}}{r}}_{\text{Rotate to }\theta_{f}} \tag{9}\]  
Figure 3: Schematic diagram of navigation guidance parameters in a two-dimensional plane.