## Optimal Control of the Second Order Elliptic Equations with Biomedical Applications

##### Abstract

Dissertation analyzes optimal control of systems with distributed parameters described
by the general boundary value problems in a bounded Lipschitz domain for the linear
second order uniformly elliptic partial differential equations (PDE) with bounded measurable coefficients. Broad class of elliptic optimal control problems under Dirichlet or
Neumann boundary conditions are considered, where the control parameter is the density of sources, and the cost functional is the L2-norm difference of the weak solution
of the elliptic problem from measurement along the boundary or subdomain. The optimal control problems are fully discretized using the method of finite differences. Two
types of discretization of the elliptic boundary value problem depending on Dirichlet
or Neumann type boundary condition are introduced. Convergence of the sequence of
finite-dimensional discrete optimal control problems both with respect to the cost functional and the control is proved. The methods of the proof are based on energy estimates
in discrete Sobolev spaces, Lax-Milgram theory, weak compactness and convergence of
interpolations of solutions of discrete elliptic problems, and delicate estimation of the
cost functional along the sequence of interpolations of the minimizers for the discrete
optimal control problems. Dissertation pursues application of the optimal control theory
of elliptic systems with distributed parameters to biomedical problem on the identification of cancerous tumor. The Inverse Electrical Impedance Tomography (EIT) problem on recovering electrical conductivity tensor and potential in the body based on the measurement of the boundary voltages on the m electrodes for a given electrode current is
analyzed. A PDE constrained optimal control framework in Besov space is developed,
where the electrical conductivity tensor and boundary voltages are control parameters,
and the cost functional is the norm difference of the boundary electrode current from the
given current pattern and boundary electrode voltages from the measurements. The state
vector is a solution of the second order elliptic PDE in divergence form with bounded
measurable coefficients under mixed Neumann/Robin type boundary condition. The
novelty of the control theoretic model is its adaptation to clinical situation when additional "voltage-to-current" measurements can increase the size of the input data from
m up to m! while keeping the size of the unknown parameters fixed. Existence of
the optimal control is established. Fréchet differentiability in the Banach-Besov spaces
framework is proved and the formula for the Frechet gradient expressed in terms of the
adjoined state vector is derived. Optimality condition is formulated, and gradient type
iterative algorithm in Hilbert-Besov spaces setting is developed. EIT optimal control
problem is fully discretized using the method of finite differences. New Sobolev-Hilbert
space is introduced, and the convergence of the sequence of finite-dimensional optimal
control problems to EIT coefficient optimal control problem is proved both with respect
to functional and control in 2- and 3-dimensional domains.