On the Control of Systems Modelled by Partial Differential Equations



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This work aims to discuss two kinds of numerical problems, namely the control of diffusion phenomenon on the surface of manifolds and the control of problems modelled by parabolic variational inequalities of the obstacle type. The manifold problems treated are the control of diffusion phenomenon on the surface of a torus and sphere in R^3. Discretization are constructed using finite element in space and an implicit integration scheme in time. The calculation of the optimal control was performed in two ways. The first approach discusses the control with mapping it to a standard domain and imposing periodic boundary condition over the mesh geometry for solving the state equation. The second approach is rather a direct approach which involves working with the mesh itself; it handles general two-dimensional manifolds in three-dimensional space. The solution of the state equation is numerically approximated using an isoparametric finite element method.

The control problem associated with the parabolic variational inequalities is transformed into a control problem with the state equation as a nonlinear parabolic equation using penalization. Adjoint equation techniques are employed to compute the optimal control for the obstacle type problem and a conjugate gradient algorithm is used to solve the non-linear minimization problem that appears. Optimality conditions for the control problem were found using perturbation analysis.



Approximate Controllability, Exact Controllability, Null Controllability