Title: An optimal control of a variable-order fractional PDE

Speaker: Wang Hong (Professor, University of South Carolina)

Time: June 5, 2023, 10:00 AM

Place: 102, Building 2, Hainayuan

Abstract: 

Integer-order diffusion partial differential equations (PDEs) were derived under the assumptions that the underlying particle movements have (i) a mean free path and (ii) a mean waiting time, which hold for the diffusive transport of solutes in homogeneous aquifers when the solute plumes were observed to have Gaussian type symmetric and exponentially decaying tails. However, field tests showed that the diffusive transport of solutes in heterogeneous aquifers often exhibit highly skewed power-law decaying tails. This explains why integer-order PDEs may recover highly oscillating coefficients when used to model transport in heterogeneous media and so making the catch of highly skewed power-law decaying behavior of the solute transport in heterogeneous media a challenging task for integer-order PDEs.

Fractional diffusion PDEs were derived assuming their solutions have power-law decaying tails, and so accurately model diffusive transport in heterogeneous aquifers. However, fractional PDEs introduce new modeling, computational and mathematical issues that are not common in the context of integer-order PDEs. In this talk we will go over these issues and focus on the discussion of modeling issues. If time permits, we will also discuss mathematical, numerical, and computational issues and report recent progress in these directions.

The variable-order optimal control encounters mathematical and numerical issues that are not common in its integer-order and constant-order fractional analogues: (i) The adjoint state equation of the variable-order Caputo time-fractional PDE turns out to be a different and more complex type of variable-order Riemann-Liouville time-fractional PDE. (ii) The coupling of the variable-order fractional state PDE and adjoint state PDE, and the variational inequality reduces the regularity of the solution to the optimal control. (iii) The numerical approximation to fractional optimal control model needs to be analyzed due to the low regularity and coupling of the model. We will prove the wellposedness and regularity of the model and an optimal-order error estimate to its numerical discretization.