PDE-constrained optimization

The following example comes from p. 20-21 of Pearson.[3] Chemotaxis is the movement of an organism in response to an external chemical stimulus. One problem of particular interest is in managing the spatial dynamics of bacteria that are subject to chemotaxis to achieve some desired result. For a cell density ${\displaystyle z(t,{\bf {x}})}$ and concentration density ${\displaystyle c(t,{\bf {x}})}$ of a chemoattractant, it is possible to formulate a boundary control problem:

$${\displaystyle \min _{z,c,u}\;{1 \over {2}}\int _{\Omega }\left[z(T,{\bf {x}})-{\widehat {z}}\right]^{2}+{\gamma _{c} \over {2}}\int _{\Omega }\left[c(T,{\bf {x}})-{\widehat {c}}\right]^{2}+{\gamma _{u} \over {2}}\int _{0}^{T}\int _{\partial \Omega }u^{2}}$$

where ${\displaystyle {\widehat {z}}}$ is the ideal cell density, ${\displaystyle {\widehat {c}}}$ is the ideal concentration density, and ${\displaystyle u}$ is the control variable. This objective function is subject to the dynamics:

$${\displaystyle {\begin{aligned}{\partial z \over {\partial t}}-D_{z}\Delta z-\alpha \nabla \cdot \left[{\nabla c \over {(1+c)^{2}}}z\right]&=0\quad {\text{in}}\quad \Omega \\{\partial c \over {\partial t}}-\Delta c+\rho c-w{z^{2} \over {1+z^{2}}}&=0\quad {\text{in}}\quad \Omega \\{\partial z \over {\partial n}}&=0\quad {\text{on}}\quad \partial \Omega \\{\partial c \over {\partial n}}+\zeta (c-u)&=0\quad {\text{on}}\quad \partial \Omega \end{aligned}}}$$

where ${\displaystyle \Delta }$ is the Laplace operator.