29 research outputs found
Optimal control for evolutionary imperfect transmission problems
We study the optimal control problem of a second order linear evolution equation defined in two-component composites with e-periodic disconnected inclusions of size e in presence of a jump of the solution on the interface that varies according to a parameter γ. In particular here the case is analyzed. The optimal control theory, introduced by Lions (Optimal Control of System Governed by Partial Differential Equations, 1971), leads us to characterize the control as the solution of a set of equations, called optimality conditions. The main result of this paper proves that the optimal control of the e-problem, which is the unique minimum point of a quadratic cost functional , converges to the optimal control of the homogenized problem with respect to a suitable limit cost functional . The main difficulties are to find the appropriate limit functional for the control of the homogenized system and to identify the limit of the controls
Monotonicity Principle in Tomography of Nonlinear Conducting Materials
We treat an inverse electrical conductivity problem which deals with the
reconstruction of nonlinear electrical conductivity starting from boundary
measurements in steady currents operations. In this framework, a key role is
played by the Monotonicity Principle, which establishes a monotonic relation
connecting the unknown material property to the (measured) Dirichlet-to-Neumann
operator (DtN). Monotonicity Principles are the foundation for a class of
non-iterative and real-time imaging methods and algorithms. In this article, we
prove that the Monotonicity Principle for the Dirichlet Energy in nonlinear
problems holds under mild assumptions. Then, we show that apart from linear and
-Laplacian cases, it is impossible to transfer this Monotonicity result from
the Dirichlet Energy to the DtN operator. To overcome this issue, we introduce
a new boundary operator, identified as an Average DtN operator.Comment: 28 pages, 6 figure
Uniform resolvent convergence for strip with fast oscillating boundary
In a planar infinite strip with a fast oscillating boundary we consider an
elliptic operator assuming that both the period and the amplitude of the
oscillations are small. On the oscillating boundary we impose Dirichlet,
Neumann or Robin boundary condition. In all cases we describe the homogenized
operator, establish the uniform resolvent convergence of the perturbed
resolvent to the homogenized one, and prove the estimates for the rate of
convergence. These results are obtained as the order of the amplitude of the
oscillations is less, equal or greater than that of the period. It is shown
that under the homogenization the type of the boundary condition can change