# Introduction

Nowadays a great attention is given to a solution of problems of global optimization and synthesis of the optimal control of dynamic systems. These problems arise during designing of planes, helicopters, spacecrafts when it is necessary to optimize characteristic parameters (e.g., weight, distance of flight, aerodynamics) and develop control systems of an object or its separate elements. The objective of this work was to develop the way of application of the interval methods of global constrained optimization to solve the problem of finding the optimal control. Interval methods possess lower computational complexity (methods work with intervals and interval vectors, i.e., processing is conducted not on points but on sets) and they are less demanding on the requirements to the problem statement (e.g., there is no necessity in convexity and differentiability of optimized function) [1]. Recently, the author has developed some deterministic and heuristic interval methods of global constrained optimization [2, 3]:

- deterministic:
- method of dichotomy of direct image,
- method of cutoff of virtual values,
- method of stochastic cutoff of virtual values,
- generalized inverse method,
- method of changing directions,
- method of stochastic pull out,

- heuristic:
- method of average path,
- method of stochastic grid,
- interval scatter search,
- interval genetic algorithm.

Each method has its software realization.

# Application

All these methods are used to solve the problem of constrained global optimization. Also they can be effectively used to synthesize optimal program control and control with full feedback. The main advantage of the developed methods is that they can be used to find the optimal control of a nonlinear dynamic system with a nonlinear efficiency criteria.

*Fig. 1. Algorithm of the synthesis of the optimal control*

Algorithm of the synthesis of the optimal control consists of several main procedures:

- transformation of continuous model into discrete model (during this procedure differential equations, which describe the behavior of an object, transforms into difference equations using the one method from the given: Euler, Euler – Cauchy, Runge – Kutta third order, Runge – Kutta fourth order , continuous time – into discrete time, efficiency criteria – into discrete analogue),
- finding the optimal control for discrete model (during this procedure a sequence of intervals, which represents optimal discrete control, is obtained by optimizing the discrete analogue of the efficiency criteria),
- restoration of discrete control to continuous control (during this procedure an interval polynomial, which represents the optimal continuous control, is built basing on the previously get sequence).

Methods were successfully tested on several applied problems:

- synthesis of the optimal control of an aircraft (e.g., of a light plane during flying tests),
- aerial refueling (finding the optimal control for a tanker aircraft),
- finding the optimal control, which realize characteristic maneuvers of a helicopter,
- synthesis of the optimal control of a spacecraft (e.g., of an artificial satellite).

# Conclusion

Interval analysis can be effectively used as a base component of methods of global constrained optimization. The developed complex of interval methods has a lot of advantages:

- there are both exact and heuristic methods, thus, there is a choice between not very fast methods, which get exact solution, and fast methods, which get suboptimal solution,
- all methods are zero order methods, thus, there is no necessity to calculate derivatives and class of optimized functions is wider,
- deterministic method were proved to get the solution on which an optimized function will reach its optimal value,
- methods are easily adjusted to fulfill user’s requirements.

Interval methods have a good perspective to expand its scope to fields of robust, adaptive control and etc. In future it is planned to develop new methods of interval optimization and construct unified interval algorithm which will analyze the problem and dynamically adapt its characteristics.

# References

- Jaulin L., Kieffer M., Didrit O., Walter E., Applied interval analysis. Springer-Verlag, London, 2001.
- Panovskiy V.N., «Interval Algorithms of Finding Optimal Control with Full Feedback of Determined Discrete Systems», All-Russian Youth Scientific Conference «Applied Scientific and Technical Problems of Modern Theory of Control of Systems and Processes», Moscow, 2012.
- Panovskiy V.N., «Interval Algorithms of Finding Optimal Control with Full Feedback of Determined Continuous Systems», the 11th International Conference «Aviation and Cosmonautics – 2012», Moscow, 2012.
- Hansen E., Walster G.W., Global Optimization Using Interval Analysis. Marcel Dekker, New York, 2004.
- Moore R.E., Interval Analysis. Prentice Hall, Englewood Cliffs, 1966.