Research Projects

This proposal focuses on the solvability and control properties of the Stochastic Magneto-hydrodynamic (SMHD) equation with noise, using fixed-point theory, optimal control, stochastic differential equations, state-dependent delays, and fractional differential equations. The initial values are smooth, but solutions appear discontinuous in the presence of Poisson jumps, making it difficult to solve. To achieve the desired solution, a suitable fixed-point theory is applied. The proposed research aims to provide an elaborate study on system models with controllability analysis and optimal control for control design. Real-life applications can be established, such as MHD power plant models with stochastic disturbances and MHD solar wind models with continuous stochastic noises. The developed theory can also be applied to extended versions of MHD equations like Hall MHD, two-fluid MHD, and electron MHD systems. Key ideas in this work include fixed point theorems and compact operator theory. Fixed-point approaches are advantageous for finding solutions in abstract space and provide a suitable convergence theory. The proposed control models have the existence of optimal control and sufficient conditions for controllability. The assumption on nonlinear functions is Lipschitz continuous, and the compactness of operators is a stronger restriction than in many practical problems. The developed solvability and controllability results on SMHD provide a novel result and an exciting application for modeling the SMHD system with solar coronal magnetic evolution with stochastic trajectories.