## Issue 182, article 4

DOI:https://doi.org/10.15407/kvt182.02.034

Kibern. vyčisl. teh., 2015, Issue 182, pp.

National Technical University of Ukraine «Kyiv Polytechnic Institute» (Kiev)

**OPTIMAL CONTROL BY VIBRATIONS OF THE BEAM WITH VARIABLE CROSS-SECTION**

**Introduction. **The last half-century is characterized by the rapid development of technology. Significant progress has been made in the rocket, aircraft, shipbuilding and space technology, etc. All sectors have oscillatory processes. In some cases, they can usefully be taken into account to improve the quality of the process, while others, on the contrary, it is necessary to suppress because of their negative impact on the final process. This means that the oscillatory processes must not only be learned, but also be able to manage them effectively. Similar problems effectively manage mechanical processes just studying optimal control theory. The purpose of this article is to study the linear-quadratic problem of optimal control by oscillations of the beam with variable cross-section in the case of the free ends of the beam.

**Statement of the Problem. **The state equation is linear partial differential equation of the fourth order of hyperbolic type with given initial conditions and homogeneous boundary conditions. Quality of the process is estimated by quadratic functional. The admissible control is such a function which belongs to the class of square Lebesgue integrable functions. Optimal control is admissible control which is implemented at least the cost functional.

**The purpose **of the paper is to determine the necessary conditions for optimal control of process vibrations of a beam of variable cross-section in the case of the free ends of the beam and to give solution of integral-differential Riccati equations for the optimal control.

**The main results. **Necessary optimality conditions for the considered optimization problem are obtained. Analysis of these conditions made it possible to bring the system of integro-differential Riccati equations with partial derivatives. The solution of this system is used in the construction of an explicit formula for the calculation of optimal control.

**Conclusions. **The article investigates the linear-quadratic optimal control process vibrations of a beam of variable cross-section in the case of the free ends of the beam. Necessary optimality conditions for the considered optimization problem are obtained. Analysis of these conditions made it possible to bring the system of integro-differential Riccati equations with partial derivatives. The solution of this system is used in the construction of an explicit formula for the calculation of optimal control. Further development of the obtained results is to study the case where the control time tends to infinity. In the theory of optimal control, this problem is called the problem of analytical construction of the regulator.

** Keywords: **linear quadratic optimal control problem, method of Lagrange multipliers, necessary optimality conditions, oscillations of the beam, partial derivatives, system of integro-differential equations.

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**Received** 10.07.2015