Project's information
| Project's title | Direct, inverse and optimal control problems for new classes of fractional differential equations |
| Project’s code | QTRU01.01/21-22 |
| Research hosting institution | Institute of Mathematics |
| Coordinating unit, co-chair | Chelyabinsk State University, Russia |
| Project leader’s name | Bui Trong Kien and Vladimir Evgenyevich Fedorov |
| Project duration | 01/06/2021 - 30/06/2023 |
| Project’s budget | 200 million VND |
| Classify | Excellent |
| Goal and objectives of the project | Focus on qualitative study for direct problems, inverse problems and optimal control problems giverned by fractional differential equations |
| Main results | Research results:
a) Initial value problems for some classes of linear evolution equation with several fractional derivatives
We proved the existence and uniqueness of solutions to the initial problems for linear inhomogeneous equations of a general form with several Gerasimov–Caputo fractional derivatives in Banach spaces. The obtained results are applied to the study of a class of initial-boundary value problems for equations with several Gerasimov–Caputo time derivatives and with polynomials with respect to a self-adjoint elliptic differential operator in space variables.
b) Distributed control for semilinear equations with Gerasimov–Caputo fractional derivatives
We consider the optimal control problem for semilinear evolution equations where the higher fractional derivative are presented via lower fractional derivatives. The operator depends on the Gerasimov–Caputo fractional derivatives of lower orders is nonlinear. We then prove the existence of an optimal control under a weaker condition of uniform in time local Lipschitz continuity with respect to the phase variables of the nonlinear operator, instead of the condition of its Lipschitz continuity.
c) Linear inverse problems for multi-term equations with Riemann — Liouville derivatives
We consider the well-posedness of linear inverse coefficient problems for
multi-term equations in Banach spaces with fractional Riemann – Liouville derivatives and with bounded operators. We then gave some criteria on the well-posedness for the problem.
d) Optimal control problem governed by fractional differential equations with control constraints
We proved the existence of optimal solutions and established first-and second-order optimality conditions for the problem. Besides, we showed that if the state equation is linear, then the optimal solutions are Holder continuous.
Education and tranining result: Nguyen Quoc Tuan who is a member of the project and a PhD student of the Institute of Mathematics, completed his doctoral thesis. He will defend his thesis in 2024.
Cooperation result: establised a good relationship on research coopertions with professor V.E. Fedorov’s group of the Department of Mathematics, Chelyabinsk State University. |
| Novelty and actuality and scientific meaningfulness of the results | Give a new result on the existence of solution to a class of evolution equation with fractional derivatives in Banach spaces |
| Products of the project | List of pulications related to the project. |
| Recommendations | The project is completed on time. The obtaned results meet demands of the objective and research contents of the project. Some research contents of the project are continued to studty in the near future. |
