## 3 Optimization-Based Strategies
### Global Information-Driven Optimization
#### 3.2.1 Pontryagin's Minimum Principle (PMP)  
The minimum principle can be expressed as: the optimal trajectory determined by the optimal control law must be the minimum value within the whole control domain when the control variables are limited to a certain range, which can be applied in the energy management for HEVs. The PMP [72; 73] method solves the optimal energy management by finding the instantaneous minimum value of Hamiltonian function at each moment, which is generally defined as the derivative of the stage cost function versus time represented by the adjustable variable as independent variable. The optimal control variables \(U\) will be obtained when the derivative of Hamiltonian function \(H\) versus control variables equals to 0, that is, \(\partial H/\partial U=0\). Therefore, [PERSON]'s minimum principle provides a necessary condition, but not a sufficient condition for optimality. When the obtained local optimal trajectory is a unique trajectory that satisfies necessary boundary conditions and constraints and the state function can be used to determine whether the results are the optimal solution, the optimal trajectory of PMP-based strategy is regarded as the global optimum solution. The essence of the PMP-based strategy for HEVs is to minimize the sum of fuel consumption at each moment under determined terminal constraints, the objective function, and the admissible control range. Within the boundary condition of control variables, the value of the Hamiltonian function, along with the optimal trajectory, is a constant; thus, the optimal control sequence can be obtained by finding the minimum value of performance index function. Thus, the performance index function can be expressed as:  
\[J(u)=\int_{t_{0}}^{t_{f}}L(x(t),u(t),t)dt=\int_{t_{0}}^{t_{f}}\dot{m}_{f}(u(t),t)dt \tag{8}\]  
where \(x\) is the state variable, \(u\) is the control variable, \(m_{f}\) is the fuel mass flow rate (g/s), \(L(\cdot)\) is the objective function, \(t\) is the time, \(t_{0}\) is the initial time of driving cycle, and \(t_{f}\) is the terminal time of driving cycle. Taking the battery SOC (or fuel rate) as the state variable and the power battery, engine speed, engine torque (or torque split ratios) as the control variable, the state equation and boundary condition of the system are formulated as follows:  
\[\dot{SOC}(t)=-\frac{U_{OC}-\sqrt{U_{OC}^{2}-4R_{itt}\cdot P_{ batt}}}{2R_{itt}\cdot C} \tag{9}\]  
\[\left\{ \begin{array}{c}SOC(t_{0})=c\\ SOC\Big{(}t_{f}\Big{)}=SOC(t_{0})\\ SOC_{min}\leq SOC(t)\leq SOC_{max}\\ P_{ batt\_min}\leq P_{batt}(t)\leq P_{batt\_max}\end{array} \right. \tag{10}\]  
where \(U_{OC}\) is the open-circuit voltage, \(R_{itt}\) is the internal resistance of the battery, \(P_{batt}\) is the electric power, \(C\) is the nominal capacity of the battery with a fixed value, \(c\) is a constant, \(SOC(\cdot)\) is the boundary values of state variable, \(P_{batt}(\cdot)\) is the boundary values of control variable, \(SOC_{min}\) is the minimum SOC value, \(SOC_{max}\) is the maximum SOC value, \(P_{batt\_min}\) is the minimum value of power battery, and \(P_{batt\_max}\) is the maximum value of the power battery. When solving the optimal problem based on the minimum principle, it is necessary to introduce the Hamiltonian function and the covariant variable. The Hamiltonian function can be defined as:  
\[H(x(t),u(t),\lambda(t),t)=\dot{m}_{f}(t,u)+\lambda(t)f(x(t),u(t),t) \tag{11}\]  
\[\dot{\lambda}=-\frac{\partial H}{\partial x} \tag{12}\]  
where \(f(\cdot)\) is the state equation of the system, \(\lambda\) is the covariant variable, which is related to the state transfer function. In the solving process, it is worth considering the complexity of the solutions to the boundary value problem. On the one hand, the common method for solving the above problem is the shooting method, which converts the boundary value problem to a problem of solving several initial values. The terminal SOC value is calculated by the state equation and the costate equation.