1. **State the problem:** We want to maximize the integral $$\max \int_0^T \left(2u - \frac{1}{2} u^2 + 2x\right) e^{-2t} dt$$ subject to the dynamic constraint $$\dot{x} = x - u$$ with initial condition $$x(0) = x_0$$ and free terminal state $$x(T)$$. 2. **Formulate the Hamiltonian:** The Hamiltonian for this optimal control problem is $$H = \left(2u - \frac{1}{2} u^2 + 2x\right) e^{-2t} + \lambda (x - u)$$ where $$\lambda(t)$$ is the costate variable. 3. **Optimality condition for control:** To maximize $$H$$ with respect to $$u$$, set $$\frac{\partial H}{\partial u} = 0$$ which gives $$\frac{\partial}{\partial u} \left( (2u - \frac{1}{2} u^2) e^{-2t} + \lambda (x - u) \right) = 0$$ 4. **Calculate the derivative:** $$\frac{\partial H}{\partial u} = (2 - u) e^{-2t} - \lambda = 0$$ 5. **Solve for control $$u$$:** $$ (2 - u) e^{-2t} = \lambda \implies 2 - u = \lambda e^{2t} \implies u = 2 - \lambda e^{2t} $$ 6. **Costate equation:** The costate dynamics are given by $$ \dot{\lambda} = -\frac{\partial H}{\partial x} $$ Calculate $$ \frac{\partial H}{\partial x} = 2 e^{-2t} + \lambda $$ so $$ \dot{\lambda} = - (2 e^{-2t} + \lambda) = -2 e^{-2t} - \lambda $$ 7. **State equation:** Recall $$ \dot{x} = x - u = x - (2 - \lambda e^{2t}) = x - 2 + \lambda e^{2t} $$ 8. **System of differential equations:** $$ \begin{cases} \dot{x} = x - 2 + \lambda e^{2t} \\ \dot{\lambda} = -2 e^{-2t} - \lambda \end{cases} $$ 9. **Boundary conditions:** $$ x(0) = x_0, \quad \lambda(T) = 0 \quad \text{(since } x(T) \text{ is free)} $$ 10. **Summary:** The optimal control is $$ u^*(t) = 2 - \lambda(t) e^{2t} $$ where $$x(t)$$ and $$\lambda(t)$$ satisfy the above system with given boundary conditions. This fully characterizes the optimal state and control variables maximizing the Hamiltonian.