Time estimation of control process of a linear dynamic system with dissipation

<p>In this paper we consider the problem of control process of a linear dynamic system with dissipation. &nbsp;Model described by the second-order differential equation. A control method is proposed which drives the controlled system into the null state in a finite time. The control on the right side of the equation is assumed to be bounded in absolute value. By bringing in the rest will understand that the deviation from the equilibrium position of the oscillator and the speed in this position are identically zero. For the dissipation term in the equation corresponds the first time derivative of the angle of deviation. The task has three formal objectives. Firstly we should find u (t) &nbsp;such that the system provides to its equilibrium position in the shortest time. Then we need to construct a switching curve ( a curve on which control changes sign ) of the dynamical system. In conclusion, the time required to evaluate the control process (for the case with different number of switching) . We used various methods of the theory of dynamic systems .Using Pontryagin&#39;s maximum principle construct optimal time control. Using the boundary conditions for the optimal arcs of curves we can build the first arc of the switching curve. Then, using an algorithm to build the next arc of switching from the previous obtain a parametric representation of &nbsp;j - th arc of the curve shift . And finally our purpose is the estimation of the time of motion along the optimal phase trajectory. In this case we suppose the initial point lies on the j-th arc of the logarithmic spiral of the switching curve. Then we should construct estimate dependent on the total number of sections counting from the origin of coordinates. The process of construction time estimates varied greatly depending on the position of the starting point of our dynamic process. In this paper, we construct estimates for the number of starts j &gt;= 2 &nbsp;, and for the case of no switching . In the future we plan to study the extension of the model characterizes the change in estimates under &nbsp;a factor responsible for the oscillation frequency .</p>