# Computational Algorithm for Optimal Control of an Object with Distributed Parameters in a Nonsmooth Area of Final States

M.Y. Livshits, A.V. Nenashev, Yu.E. PleshivtsevaWe propose the effective computational algorithm for solving boundary-value problems of time-optimal and maximum accuracy control with a minimax estimation of the deviation of the final trajectory from a given state. The problem is reduced to a nonconvex nonlinear programming problem. The proposed algorithm takes into account the non-convex nature of the problem of nonlinear programming, provides a search in the "ravines" zone, performs a search quite efficiently under conditions of increased dimension of the definition domain of the optimized functional, and provides the required accuracy of the solution. Due to the transformation of the multidimensional non-convex nonlinear programming problem to the problem of minimizing a smooth monotonically decreasing function of one variable, the algorithm significantly reduces the computational complexity of solving boundary-value problems of optimal speed and maximum accuracy with a minimax estimate of the deviation of the final trajectory from a given state. We give an example of the solution of the test optimal control problem for induction heating of a cylindrical billet.Full text

- Keywords
- distributed parameters; boundary-value problem; optimality criterion; search procedure; local minimum; global minimum.
- References
- 1. Rapoport E.Ya. Alternansny metod v prikladnikh zadachakh optimizatsii [An Alternative Method in Applied Optimization Problems]. Moscow, Nauka, 2000. (in Russian)

2. Livshits M.Yu., Nenashev A.V. Effective Computational Procedure of the Alternance Optimization Method. Journal of Samara State Technical University, Physical and Mathematical Science, 2019, vol. 23, no. 2, pp. 361-377. (in Russian)

3. Chistyakova T.B., Boykova O.G., Chistyakov A.N. Intellektual'noe upravlenie mnogoassortimentnym koksokhimicheskim proizvodstvom [Intelligent Control of the Multi-Assortment Coke-Chemical Production]. St. Petersburg, Profession, 2010. (in Russian)

4. Chistyakov A.N., Chistyakova T.B., Boykova O.G. Integrated Intelligent Structures for Controlling the Process of Coking. Coke and Chemistry, 1998, no. 8, pp. 18-22.(in Russian)

5. Chistyakova T.B., Kudlay V.A., Novozhilova I.V., Suvorov S.A., Kozlov V.V. Decision Support System for Service of Refractory Lining of Steelmaking Converter. Bulletin of the Saint Petersburg State Institute of Technology, 2016, no. 37, pp. 60-66. (in Russian)

6. Ostrovskiy Yu.V., Chistyakova T.B., Malin A.A. System for Controlling the Production of Substances of Medical Preparations with Variable Technology. Chemical Industry, 2003, vol. 80, no. 5, pp. 38-43. (in Russian)

7. Chistyakova T.B., Shlyago Yu.I., Novozhilova I.V., Mal'tseva N.V. Intellektual'nye sistemy tekhnologicheskogo proektirovaniya, upravleniya i obucheniya v mnogoassortimentnom proizvodstve granulirovannykh porisykh materialov iz tonkodispersnykh chastits [Intelligent Systems of Technological Design, Control and Training for the Multi-Assortment Production of Granular Porous Materials of Fine Particles]. St. Petersburg, Saint Petersburg State Institute of Technology, 2012. (in Russian)

8. Livshitc M.Yu., Sizikov A.P. Multi-Criteria Optimization of Refinery. Thermophysical Basis of Energy Technologies, 2015, vol. 110, article ID: 01035. DOI 10.1051/epjconf/201611001035

9. Livshits M.Yu. System Optimization of Processes of Heat and Mass Transfer of Technological Thermophysics. Mathematical Methods in Engineering and Technology, 2016, vol. 11, pp. 104-114. (in Russian)

10. Polosin A.N., Chistyakova T.B. Mathematical Model of Single-Screw Extrusion for Control of Plastic Material Quality in Multi-Assortment Productions of Polymeric Films. Sistemy upravleniya i informatsionnye tekhnologii [Control Systems and Information Technologies], 2009, no. 2, pp. 87-92. (in Russian)

11. Vasilyev F.P. Chislennie metodi resheniya ekstremalnikh zadach [Numerical Methods for Solving Extremal Problems]. Moscow, Nauka, 1980. (in Russian)

12. Rapoport E.Ya., Pleshivtseva Yu.E. Optimalnoe upravlenie temperaturnymi rezhimami induktsionnogo nagreva [Optimum Control of Temperature Modes of Induction Heating]. Moscow, Nauka, 2012. (in Russian)

13. Egorov Yu.V. Necessary Conditions for Optimality of Control in a Banach Space. Mathematical Collection, 1964, vol. 64 (106), no. 1, pp. 79-101.(in Russian)

14. Butkovskiy A.G. Teoriia optimal'nogo upravleniia sistemami s raspredelennymi parametrami [Theory of Optimal Control of Systems with Distributed Parameters]. Moscow, Nauka, 1965. (in Russian)

15. Gill P., Murray W., Wright M. Practical Optimization. N.Y., Academic Press, 1981.

16. Samarskii A.A. Vvedenie v chislennye metody [An Introduction to Numerical Methods]. Moscow, Nauka, 1997.(in Russian)

17. Zakharova E.M., Minashina I.K. Review of Multidimensional Optimization Techniques. Informatsionnye Protsessy, 2014, vol. 14, no. 3, pp. 256-274.(in Russian)

18. Chichinadze V.K. Reshenie nelineynykh nevypuklykh zadach optimizatsii [Solution of Nonconvex Nonlinear Optimization Problems]. Moscow, Nauka, 1983. (in Russian)

19. Nelder J.A., Mead R.A. Simplex Method for Function Minimization. Computer Journal, 1965, vol. 7, pp. 308-313.