Optimal control problem for quasilinear hyperbolic system:
the Slutsky equation

T. O. Derevianko; V. M. Kyrylych

doi:10.15330/ms.43.1.66-77

A model for determination the change of demand for goods at the change of price on commodities and capital of consumer is constructed. This model is described by a system of quasi-linear hyperbolic, which is written in Riemann invariants. Using the results of theory of ordinary differential equations with parameters we prove the existence of a classical solution to mixed hyperbolic problem. Also applying the method of linearization and non-classical internal variation the necessary optimality conditions for finding a given level of demand at the change of price on commodities and capital of consumer are shown.

1. Ashmanov S.A. Mathematical models and methods in economics. – I.: Moscow University, 1980. – 199 p. (in Russian)

2. Arguchintsev A.V. Optimal control of hyperbolic systems. – M.: FIZMATLIT, 2007. – 168 p. (in Russian)

3. Gugat M. Optimal nodal control of networked hyperbolic systems: evaluation of derivatives// Advanced Modeling and Optimization. – 2005. – V.7, №1. – P. 9–37.

4. Matveev H.I., Yakubovych V.A., Optimal control systems: Ordinary differential equations. Special problems. – Uzd-vo S.-Peterburg., 2003. – 540 p. (in Russian)

5. Erofeenko V.T., Kozlovskaya I.S., Partial differential equations and mathematical models in economics: Course of lectures. Uzd-vo 2, revised and complemented. – I.: Edytoral URSS, 2004. – 248 p. (in Russian)

6. Syharev A.G., Tymohov A.V., Fedorov V.V. The course of optimization methods. – I.: Nayka, GRFML, 1986. – 328 p. (in Russian)

7. Myshkis A.D., Filimonov A.I. Continuous solution of quasilinear hyperbolic system of equations with two independent variables// Differential Equations. – 1981. – V.17, №3. – P. 488–500. (in Russian)

8. Ptashnyk B.I., Ilkiv V.S., Kmit I.Ya., Polishchuk V.M., Nonlocal Boundary Value Problems for Partial Differential Equations, Kiev: Naukova Dumka, 2002. – 415 p. (in Ukrainian)

9. Myshkis A.D., Filimonov A.I. On the global continuous solvability of the mixed problem for onedimensional hyperbolic systems of quasilinear equations// Differential Equations. – 2008. – V.44, №3. – P. 394–407. (in Russian)

10. Filimonov A.M. Sufficient conditions of global solvability of the mixed problem for the quasilinear hyperbolic systems with two independent variables. – M.: 1980, 17 p. – Dep. in VINITI 19.12.80, №6–81. (in Russian)

11. D’Acunto B. Frunzo L. Qualitative analysis and simulations of a free boundary problem for multispecies biofilm models// Mathematical and Computer Modeling. – 2011. – V.53. – P. 1596–1606.

12. D’Acunto B., Frunzo L. Free boundary problem for an initial cell layer in multispecies biofilm formation// Applied Mathematics Letters. – 2012. – V.25. – P. 20–26.

13. Samojlenko A.M., Perestjyk M.O., Parasjyk I.O., Differential equation. – Kyiv: Lubid, 2003. – 600 p. (in Ukrainian)

14. Fichtenholz G.M., A course of differential and integral calculus. – Moscow: Nauka, 1969, V.II. – 800 p. (in Russian)

15. Moklyachuk M.P., Variation calculation. Extreme problems. – Kyiv, 2004, 384 p. (in Ukrainian)

	Optimal control problem for quasilinear hyperbolic system: the Slutsky equation (in Ukrainian)
Author	T. O. Derevianko, V. M. Kyrylych taras_derevianko@ukr.net Ivan Franko National University of Lviv
Abstract	A model for determination the change of demand for goods at the change of price on commodities and capital of consumer is constructed. This model is described by a system of quasi-linear hyperbolic, which is written in Riemann invariants. Using the results of theory of ordinary differential equations with parameters we prove the existence of a classical solution to mixed hyperbolic problem. Also applying the method of linearization and non-classical internal variation the necessary optimality conditions for finding a given level of demand at the change of price on commodities and capital of consumer are shown.
Keywords	Slutsky demand; quasilinear hyperbolic system; necessary optimality conditions
DOI	doi:10.15330/ms.43.1.66-77
Reference	1. Ashmanov S.A. Mathematical models and methods in economics. – I.: Moscow University, 1980. – 199 p. (in Russian) 2. Arguchintsev A.V. Optimal control of hyperbolic systems. – M.: FIZMATLIT, 2007. – 168 p. (in Russian) 3. Gugat M. Optimal nodal control of networked hyperbolic systems: evaluation of derivatives// Advanced Modeling and Optimization. – 2005. – V.7, №1. – P. 9–37. 4. Matveev H.I., Yakubovych V.A., Optimal control systems: Ordinary differential equations. Special problems. – Uzd-vo S.-Peterburg., 2003. – 540 p. (in Russian) 5. Erofeenko V.T., Kozlovskaya I.S., Partial differential equations and mathematical models in economics: Course of lectures. Uzd-vo 2, revised and complemented. – I.: Edytoral URSS, 2004. – 248 p. (in Russian) 6. Syharev A.G., Tymohov A.V., Fedorov V.V. The course of optimization methods. – I.: Nayka, GRFML, 1986. – 328 p. (in Russian) 7. Myshkis A.D., Filimonov A.I. Continuous solution of quasilinear hyperbolic system of equations with two independent variables// Differential Equations. – 1981. – V.17, №3. – P. 488–500. (in Russian) 8. Ptashnyk B.I., Ilkiv V.S., Kmit I.Ya., Polishchuk V.M., Nonlocal Boundary Value Problems for Partial Differential Equations, Kiev: Naukova Dumka, 2002. – 415 p. (in Ukrainian) 9. Myshkis A.D., Filimonov A.I. On the global continuous solvability of the mixed problem for onedimensional hyperbolic systems of quasilinear equations// Differential Equations. – 2008. – V.44, №3. – P. 394–407. (in Russian) 10. Filimonov A.M. Sufficient conditions of global solvability of the mixed problem for the quasilinear hyperbolic systems with two independent variables. – M.: 1980, 17 p. – Dep. in VINITI 19.12.80, №6–81. (in Russian) 11. D’Acunto B. Frunzo L. Qualitative analysis and simulations of a free boundary problem for multispecies biofilm models// Mathematical and Computer Modeling. – 2011. – V.53. – P. 1596–1606. 12. D’Acunto B., Frunzo L. Free boundary problem for an initial cell layer in multispecies biofilm formation// Applied Mathematics Letters. – 2012. – V.25. – P. 20–26. 13. Samojlenko A.M., Perestjyk M.O., Parasjyk I.O., Differential equation. – Kyiv: Lubid, 2003. – 600 p. (in Ukrainian) 14. Fichtenholz G.M., A course of differential and integral calculus. – Moscow: Nauka, 1969, V.II. – 800 p. (in Russian) 15. Moklyachuk M.P., Variation calculation. Extreme problems. – Kyiv, 2004, 384 p. (in Ukrainian)
Pages	66-77
Volume	43
Issue	1
Year	2015
Journal	Matematychni Studii
Full text of paper	pdf
Table of content of issue	html

Optimal control problem for quasilinear hyperbolic system: the Slutsky equation (in Ukrainian)