Research Article
Optimal Control Problem for Fourth-Order Bianchi Equation in Variable Exponent Sobolev Spaces
Abstract
This work proposes a necessary and sufficient condition such as Pontryagin’s maximum principle for
an optimal control problem with distributed parameters, which is described by the fourth-order Bianchi equation involving coefficients in variable exponent Lebesgue spaces. The problem is studied by aid of a novel version of the increment method that essentially uses the concept of the adjoint equation of integral type.
Keywords
4d optimal control Pontryagin's maximum principle Bianchi equation variable exponent Lebesgue spaces variable exponent Sobolev spaces
References
- Akhiev, S.S., The general form of linear bounded functionals in an anisotropic space of S. L. Sobolev type, Doklady Akademii Nauk Azerbaijan SSR, 35(6)(1979), 3–7 (in Russian).
- Akhiev, S.S., Fundamental solutions of some local and nonlocal boundary value problems and their representations, Doklady Akademii Nauk SSSR, 271(2)(1983), 265–269 (in Russian).
- Akhmedov, K.T., Akhiev, S.S., Necessary conditions for optimality for some problems in optimal control theory, Doklady Akademii Nauk Azerbaijan SSR, 28(5) (1972), 12–16 (in Russian).
- Aramaki, J., An extension of a variational inequality in the Simader theorem to a variable exponent Sobolev space and applications: the Dirichlet case, International Journal of Analysis and Applications, 20(2022), Article number 13.
- Bandaliyev, R.A., Guliyev, V.S., Mamedov, I.G., Rustamov, Y.I., Optimal control problem for Bianchi equation in variable exponent Sobolev spaces, Journal of Optimization Theory and Applications, 180(1)(2019), 303–320.
- Bandaliyev, R.A., Guliyev, V.S., Mamedov, I.G., Sadigov, A.B., The optimal control problem in the processes described by the Goursat problem for a hyperbolic equation in variable exponent Sobolev spaces with dominating mixed derivatives, Journal of Computational and Applied Mathematics, 305(2016), 11–17.
- Bandaliyev, R.A., Mamedov, I.G., Mardanov, M.J., Melikov, T.K., Fractional optimal control problem for ordinary differential equation in weighted Lebesgue spaces, Optimization Letters, 14(6)(2020), 1519–1532.
- Bianchi, L., Sulla estensione del metodo di Riemann alle equazioni lineari alle derivate parziali d’ordine superiore, Accademia dei Lincei Rendiconti Serie V Classe di Scienze Fisiche, Matematiche e Naturali, 4(1895), 89–99 (in Italian).
- Bonino, B., Estatico, C., Lazzaretti, M., Dual descent regularization algorithms in variable exponent Lebesgue spaces for imaging, Numerical Algorithms, 92(1)(2023), 149–182.
- Cruz-Uribe, D.V., Fiorenza, A., Variable Lebesgue Spaces: Foundations and Harmonic Analysis, in: Series Applied and Numerical Harmonic Analysis, Birkhauser, 2013.
- Çiçek, G., Mahmudov, E.N., Optimization of Mayer functional in problems with discrete and differential inclusions and viability constraints, Turkish Journal of Mathematics, 45(5)(2021), 2084–2102.
- Demir Sağlam, S., Mahmudov, E.N., On duality in convex optimization of second-order differential inclusions with periodic boundary conditions, Hacettepe Journal of Mathematics and Statistics, 51(6)(2022), 1588–1599.
- Diening, L., Harjulehto, P., Hasto, P., Ruzicka, M., Lebesgue and Sobolev Spaces with Variable Exponents, in: Series Lecture Notes in Mathematics 2017, Springer-Verlag, Heidelberg, 2011.
- Gamkrelidze, R.V., Principles of Optimal Control Theory, Springer, New York, 1978.
- Koshcheeva, O.A., Construction of the Riemann function for the Bianchi equation in an n-dimensional space, Russian Mathematics, 52(9)(2008), 35–40.
- Mahmudov, E.N., On duality in problems of optimal control described by convex differential inclusions of Goursat-Darboux type, Journal of Mathematical Analysis and Applications, 307(2)(2005), 628–640.
- Mahmudov, E.N., Duality in optimal control with first order partial differential inclusions, Applicable Analysis, 102(4)(2023), 1161–1182.
- Mahmudov, E.N., Optimal control of differential inclusions with endpoint constraints and duality, Applicable Analysis, (2022).
- Mahmudov, E.N., Çiçek, G., Optimization of differential inclusions of Bolza type and dualities, Applied and Computational Mathematics, 6(1)(2007), 88–96.
- Mahmudov, E.N., Demir Sa˘glam, S., Necessary and sufficient conditions of optimality for second order discrete and differential inequalities, Georgian Mathematical Journal, 29(3)(2022), 407–424.
- Mahmudov, E.N., Mardanov, M.J., On duality in optimal control problems with second-order differential inclusions and initial-point constraints, Proceedings of The Institute of Mathematics and Mechanics, 46(1)(2020), 115–128.
- Mardanov, M.J., Melikov, T.K., Malik, S.T., Malikov, K., First- and second-order necessary conditions with respect to components for discrete optimal control problems, Journal of Computational and Applied Mathematics, 364(2020), Article number 112342.
- Mironov, A.N., Classes of Bianchi equations of third order, Mathematical Notes, 94(3-4)(2013), 369–378.
- Mironov, A.N., On some classes of fourth-order Bianchi equations with constant ratios of Laplace invariants, Differential Equations, 49(12)(2013), 1524–1533.
- Mironov, A.N., Darboux problem for the third-order Bianchi equation, Mathematical Notes, 102(1-2)(2017), 53–59.
- Mironov, A.N., Construction of the Riemann-Hadamard function for the three-dimensional Bianchi equation, Russian Mathematics, 65(3)(2021), 68–74.
- Mironov, A.N., Darboux problem for the fourth-order Bianchi equation, Differential Equations, 57(3)(2021), 328–341.
- Mironov, A.N., Yakovleva, Y.O., Constructing the Riemann-Hadamard function for a fourth-order Bianchi equation, Differential Equations, 57(9)(2021), 1142–1149.
- Pontryagin, L.S., Boltyanskii, V.G., Gamkrelidze, R.V., Mishenko, E.F., Mathematical Theory of Optimal Processes, Nauka, Moscow, 1969 (in Russian).
- Soltanov, K.N., On semi-continuous mappings, equations and inclusions in a Banach space, Hacettepe Journal of Mathematics and Statistics, 37(1)(2000), 9–24.
- Vasilev, F.P., Methods for Solving Extremal Problems, Nauka, Moscow, 1965 (in Russian).
- Vasilev, F.P., Optimization Methods, Faktorial Press, Moscow, 2002 (in Russian).
Abstract
References
- Akhiev, S.S., The general form of linear bounded functionals in an anisotropic space of S. L. Sobolev type, Doklady Akademii Nauk Azerbaijan SSR, 35(6)(1979), 3–7 (in Russian).
- Akhiev, S.S., Fundamental solutions of some local and nonlocal boundary value problems and their representations, Doklady Akademii Nauk SSSR, 271(2)(1983), 265–269 (in Russian).
- Akhmedov, K.T., Akhiev, S.S., Necessary conditions for optimality for some problems in optimal control theory, Doklady Akademii Nauk Azerbaijan SSR, 28(5) (1972), 12–16 (in Russian).
- Aramaki, J., An extension of a variational inequality in the Simader theorem to a variable exponent Sobolev space and applications: the Dirichlet case, International Journal of Analysis and Applications, 20(2022), Article number 13.
- Bandaliyev, R.A., Guliyev, V.S., Mamedov, I.G., Rustamov, Y.I., Optimal control problem for Bianchi equation in variable exponent Sobolev spaces, Journal of Optimization Theory and Applications, 180(1)(2019), 303–320.
- Bandaliyev, R.A., Guliyev, V.S., Mamedov, I.G., Sadigov, A.B., The optimal control problem in the processes described by the Goursat problem for a hyperbolic equation in variable exponent Sobolev spaces with dominating mixed derivatives, Journal of Computational and Applied Mathematics, 305(2016), 11–17.
- Bandaliyev, R.A., Mamedov, I.G., Mardanov, M.J., Melikov, T.K., Fractional optimal control problem for ordinary differential equation in weighted Lebesgue spaces, Optimization Letters, 14(6)(2020), 1519–1532.
- Bianchi, L., Sulla estensione del metodo di Riemann alle equazioni lineari alle derivate parziali d’ordine superiore, Accademia dei Lincei Rendiconti Serie V Classe di Scienze Fisiche, Matematiche e Naturali, 4(1895), 89–99 (in Italian).
- Bonino, B., Estatico, C., Lazzaretti, M., Dual descent regularization algorithms in variable exponent Lebesgue spaces for imaging, Numerical Algorithms, 92(1)(2023), 149–182.
- Cruz-Uribe, D.V., Fiorenza, A., Variable Lebesgue Spaces: Foundations and Harmonic Analysis, in: Series Applied and Numerical Harmonic Analysis, Birkhauser, 2013.
- Çiçek, G., Mahmudov, E.N., Optimization of Mayer functional in problems with discrete and differential inclusions and viability constraints, Turkish Journal of Mathematics, 45(5)(2021), 2084–2102.
- Demir Sağlam, S., Mahmudov, E.N., On duality in convex optimization of second-order differential inclusions with periodic boundary conditions, Hacettepe Journal of Mathematics and Statistics, 51(6)(2022), 1588–1599.
- Diening, L., Harjulehto, P., Hasto, P., Ruzicka, M., Lebesgue and Sobolev Spaces with Variable Exponents, in: Series Lecture Notes in Mathematics 2017, Springer-Verlag, Heidelberg, 2011.
- Gamkrelidze, R.V., Principles of Optimal Control Theory, Springer, New York, 1978.
- Koshcheeva, O.A., Construction of the Riemann function for the Bianchi equation in an n-dimensional space, Russian Mathematics, 52(9)(2008), 35–40.
- Mahmudov, E.N., On duality in problems of optimal control described by convex differential inclusions of Goursat-Darboux type, Journal of Mathematical Analysis and Applications, 307(2)(2005), 628–640.
- Mahmudov, E.N., Duality in optimal control with first order partial differential inclusions, Applicable Analysis, 102(4)(2023), 1161–1182.
- Mahmudov, E.N., Optimal control of differential inclusions with endpoint constraints and duality, Applicable Analysis, (2022).
- Mahmudov, E.N., Çiçek, G., Optimization of differential inclusions of Bolza type and dualities, Applied and Computational Mathematics, 6(1)(2007), 88–96.
- Mahmudov, E.N., Demir Sa˘glam, S., Necessary and sufficient conditions of optimality for second order discrete and differential inequalities, Georgian Mathematical Journal, 29(3)(2022), 407–424.
- Mahmudov, E.N., Mardanov, M.J., On duality in optimal control problems with second-order differential inclusions and initial-point constraints, Proceedings of The Institute of Mathematics and Mechanics, 46(1)(2020), 115–128.
- Mardanov, M.J., Melikov, T.K., Malik, S.T., Malikov, K., First- and second-order necessary conditions with respect to components for discrete optimal control problems, Journal of Computational and Applied Mathematics, 364(2020), Article number 112342.
- Mironov, A.N., Classes of Bianchi equations of third order, Mathematical Notes, 94(3-4)(2013), 369–378.
- Mironov, A.N., On some classes of fourth-order Bianchi equations with constant ratios of Laplace invariants, Differential Equations, 49(12)(2013), 1524–1533.
- Mironov, A.N., Darboux problem for the third-order Bianchi equation, Mathematical Notes, 102(1-2)(2017), 53–59.
- Mironov, A.N., Construction of the Riemann-Hadamard function for the three-dimensional Bianchi equation, Russian Mathematics, 65(3)(2021), 68–74.
- Mironov, A.N., Darboux problem for the fourth-order Bianchi equation, Differential Equations, 57(3)(2021), 328–341.
- Mironov, A.N., Yakovleva, Y.O., Constructing the Riemann-Hadamard function for a fourth-order Bianchi equation, Differential Equations, 57(9)(2021), 1142–1149.
- Pontryagin, L.S., Boltyanskii, V.G., Gamkrelidze, R.V., Mishenko, E.F., Mathematical Theory of Optimal Processes, Nauka, Moscow, 1969 (in Russian).
- Soltanov, K.N., On semi-continuous mappings, equations and inclusions in a Banach space, Hacettepe Journal of Mathematics and Statistics, 37(1)(2000), 9–24.
- Vasilev, F.P., Methods for Solving Extremal Problems, Nauka, Moscow, 1965 (in Russian).
- Vasilev, F.P., Optimization Methods, Faktorial Press, Moscow, 2002 (in Russian).
There are 32 citations in total.
Details
| Primary Language | English |
|---|---|
| Subjects | Calculus of Variations, Mathematical Aspects of Systems Theory and Control Theory |
| Journal Section | Articles |
| Authors | |
| Publication Date | June 30, 2024 |
| Published in Issue | Year 2024 Volume: 16 Issue: 1 |