## 2 Materials and Methods
### Open Loop Optimum Controller  
This section derives the optimum control from first principles: standard articulation of an optimization problem. While, subsequent derivation of innovative control architectures uses the optimum control developed in this section, this explanation will not be repeated in subsequent discussions. The optimum controller uses optimization theory to address the key control issue of cost. The optimum controller is defined as the one that minimizes the quadratic control cost function. To find an optimum controller, we choose to optimize the double integrator cost \(I\) and the square of the control as a function of all variables annotated \((\cdot)\) as follows. \[\text{Minimize }J[\theta(\cdot),\omega(\cdot),u(\cdot)]=\frac{1}{2}\int_{l_{0} }^{t_{f}}u^{2}dt. \tag{1}\]  
\[\text{Subject to }\dot{\theta}=\omega \tag{2}\]\[\dot{\omega}=u \tag{3}\]  
\[\dot{\theta}=\omega \tag{4}\]  
\[(\theta_{0},\omega_{0})=(0,0) \tag{5}\]  
\[\left(\theta_{f}-1,\omega_{f}-1,t_{f}-1\right)=(0,0,0) \tag{6}\]  
Make some important observations. First, notice the notation. \(\theta(\cdot),\omega(\cdot),u(\cdot)\) live in infinite dimensional functional space, while \(\theta(t),\omega(t),u(t)\) live in finite dimensional space. In addition, the smooth terminal manifold is the following: \(e\left(\theta_{f},t_{f}\right)=0\). We intend to minimize the control Hamiltonian. 1. Write the control Hamiltonian. \[H(\lambda,\theta,u,t)=F(\theta,u,t)+\lambda^{T}f(\theta,u,t)\] (7)
2. Implement the Hamiltonian Minimization Condition for the static problem of Equation (8). \[\text{min}H(\dots;u)\text{ {where }}u\in R,\text{ {that is }}\frac{\partial H}{ \partial u}=0\] (8) This is a constrained minimization problem, so use Equations (9) and (10) where \(\lambda_{\omega}\) is the Lagrange multiplier associated with the co-state. \[\frac{\partial H(\dots;u)}{\partial u}=u+\lambda_{\omega}=0\] (9) \[u(\lambda_{\omega})=-\lambda_{\omega}\] (10) Confirm optimality by verifying the convexity condition in Equation (11). \[\text{Subject to }\dot{\theta}=\omega \tag{2}\]0 (11) The result: once we find the co-state, we will have optimum control. Notice Karush-Kuhn-Tucker conditions often used with inequality constraints are not necessary here. 3. Apply the Adjoint Equations per Equations (12) and (14), which result in Equations (13) and (15), respectively, and are plotted in Figure 1b. \[\text{Subject to }\dot{\theta}=\omega \tag{2}\]1 (12) \[\text{Subject to }\dot{\theta}=\omega \tag{2}\]2 (13) \[\text{Subject to }\dot{\theta}=\omega \tag{2}\]3 (14) Integrating (14) \[\text{Subject to }\dot{\theta}=\omega \tag{2}\]4 (15)
4. Rewrite the Hamiltonian Minimization Condition in Equation (8) by substituting Equation (9). \[\text{Subject to }\dot{\theta}=\omega \tag{2}\]5 (16)Substituting Equation (15) into Equation (16) produces Equation (17). \[\text{Subject to }\dot{\theta}=\omega \tag{2}\]6  
6. Implement in SIMULINK as displayed in Figure 2 using Equations (3), (18)-(20), which are simulated per Figure 2 and whose results are plotted in Figure 1a. \[\text{Subject to }\dot{\theta}=\omega \tag{2}\]7  
\[\text{Subject to }\dot{\theta}=\omega \tag{2}\]8  
\[\text{Subject to }\dot{\theta}=\omega \tag{2}\]9  
Equation (17) will be the enforced form of the control in subsequent derivations substantiating the novel paradigm contribution for each subsequent form. Using the initial conditions \(\theta(0)=0,\omega(0)=0\) and the final conditions \(\theta(1)=1,\,\omega(1)=0\) yields the values for constants: \(a=-12,\,b=6,\,c=0\), and \(d=0.\) Thus, the optimum open loop controller is Equation (21) implemented in SIMULINK, as depicted in \[\dot{\omega}=u \tag{3}\]0  
\[\dot{\omega}=u \tag{3}\]1  
It is important to remember that \(u=-12t+6\) is not the global optimum for the simple double integrator. We established the values \(a=-12,\) and \(b=6\) using the boundary values stated, which makes this a local optimum control from \(0\leq t\leq 1\). The more general \(u=at+b\) is the global optimum control.