## 2 Optimal Control of a Multilayer Electroelastic Engine
### Characteristics of a Multilayer Electroelastic Engine with a Longitudinal Piezoeffect
In this optimal control system, the integrator transfer function \(W_{i}(s)\) has the form  
\[\frac{\xi(t)}{l}=d_{33}E_{3}(t)-\frac{s_{33}^{E}C_{e}\xi(t)}{S_{0}} \tag{2}\]0  
\[\frac{\xi(t)}{l}=d_{33}E_{3}(t)-\frac{s_{33}^{E}C_{e}\xi(t)}{S_{0}} \tag{2}\]1  
where \(k_{i}\) is the transfer coefficient of the integrator. Accordingly, for the control system with the integrator for the multilayer electroelastic engine with a longitudinal piezoelectric, considering the electrical constant time of the multilayer electroelastic engine and the transfer coefficient of the integrator, the ordinary second-order differential equation is written as  
\[\frac{\xi(t)}{l}=d_{33}E_{3}(t)-\frac{s_{33}^{E}C_{e}\xi(t)}{S_{0}} \tag{2}\]2  
\[\frac{\xi(t)}{l}=d_{33}E_{3}(t)-\frac{s_{33}^{E}C_{e}\xi(t)}{S_{0}} \tag{2}\]3  
where \(\xi(t)\), \(u(t)\), and \(k_{33}^{U}\) are the displacement, the voltage, and transfer coefficient of the control system of the multilayer electroelastic engine with a longitudinal piezoelectric, respectively. For the ordinary second-order differential Equation (6), therefore, the system of differential equations in Cauchy form is obtained,  
\[\left\{ \begin{array}{l}\frac{d\xi_{1}}{dt}=\xi_{2}\\ \frac{d\xi_{2}}{dt}=-\frac{1}{T_{n}}\xi_{2}+\frac{k_{33}^{U}}{T_{n}}u\\ \end{array} \right. \tag{8}\]  
where \(\xi_{1}=\xi\) and \(\xi_{2}=\dot{\xi}_{1}\) as the state variables are selected the displacement and the velocity of the multilayer electroelastic engine with a longitudinal piezoeffect. Application of Pontryagin Maximum Principle for Decision Optimal Control of a Multilayer Electroelastic Engine  
Using the Pontryagin maximum principle [18], the control law ensures the transition of the multilayer electroelastic engine from any initial state to a given end point at a minimum time. The end point is the origin of coordinates. The Hamilton function for this maximum principle has the following form  
\[\max_{u}H=\max_{u}\left(\psi_{1}\xi_{2}+\psi_{2}\left(-\frac{1}{T_{n}}\xi_{2}+ \frac{k_{33}^{U}}{T_{n}}u\right)\right) \tag{9}\]  
where \(\psi_{1}\) and \(\psi_{2}\) are the functions. Therefore, for the Hamilton function [19], the system of equations has the form  
\[\left\{ \begin{array}{l}\frac{d\psi_{1}}{dt}=-\frac{\partial H}{\partial\xi_ {1}}=0\\ \frac{d\psi_{2}}{dt}=-\frac{\partial H}{\partial\xi_{2}}=-\psi_{1}+\frac{1}{T_ {n}}\psi_{2}\end{array} \right. \tag{10}\]  
The solution to this system of equations has the form  
\[\left\{ \begin{array}{l}\psi_{1}=A_{1}=\text{const}\\ \psi_{2}=-A_{1}t+A_{2}e^{\frac{t}{t_{2}}}\end{array} \right. \tag{11}\]  
Therefore, the function \(\psi_{2}\) changes once the sign (see Figure 2), since the straight line and the exponent intersect once. According to the maximum principle, from (9) the control signal has two values, which equal in absolute value and opposite in the sign. The sign of the control signal coincides with the sign of this function \(\psi_{2}\) on Figure 2. The control function \(u\) has the form  
\[u=u_{0}\text{ sign }\psi_{2} \tag{12}\]  
where \(u_{0}\) is the control signal amplitude. In this optimal system with an ordinary second-order differential equation, in Figure 2, \(u\) takes only two values, \(u_{0}\) and \(-u_{0}\), and changes once. The trajectory of the control system of the multilayer electroelastic engine with a longitudinal piezoelectric in the phase plane consists of two sections, the initial and the final, with the latter passing through the origin and acts as the switching line. The switching line of the multilayer electroelastic engine is defined on the phase plane. Figure 2: The control function in the optimal system. The solution for the system of Equation (8) for \(u=\) const has the form  
\[\left\{ \begin{array}{l}\xi_{1}=k_{33}^{LI}ut-T_{n}B_{2}e^{-\frac{t}{T_{n}}}+B_{1 }\\ \xi_{2}=k_{33}^{LI}u+B_{2}e^{-\frac{t}{T_{n}}}\end{array} \right., \tag{13}\]  
when the initial location of the system is at the origin and when the time is \(\tau=-t\), the constants are found in the form  
\[\left\{