  This paper is devoted to continuous and discrete adjoint-based optimization
approaches for optimal control problems governed by an important class of
Nonlinear Coupled Anisotropic Convection-Diffusion Chemotaxis-type System
(NCACDCS). This study is motivated by the fact that the considered complex
systems (with complex geometries) appear in diverse biochemical, biological and
biosocial criminology problems. To solve numerically the corresponding
nonlinear optimization problems, the primal problem NCACDCS is discretised by a
coupled Lattice Boltzmann Method with a general Multiple-Relaxation-Time
collision operators (MRT) while for the adjoint problem, an Adjoint
Multiple-Relaxation-Time lattice Boltzmann model (AMRT) is proposed and
investigated.
  First, the optimal control problems are formulated and first-order necessary
optimality conditions are established by using sensitivity and adjoint
calculus. The resulting problems are discretised by the coupled MRT and AMRT
models and solved via gradient descent methods. First of all, an efficient and
stable modified MRT model for NCACDCS is developed, and through the
Chapman-Enskog analysis we show that NCACDCS can be correctly recovered from
the proposed MRT model. For the adjoint problem, the discretisation strategy is
based on AMRT model, which is found to be as simple as MRT model with also
highly-efficient parallel nature. The derivation of AMRT model and the discrete
cost functional gradient are derived mathematically in detail using the
developed MRT model. The obtained method is reliable, efficient, practical to
implement and can be easily incorporated into any existing MRT code.
