An optimal control problem when controls act on the boundary can also be understood as a variational principle under differential constraints and no restrictions on boundary and/or initial values. From this perspective, some existence theorems can be proved when cost functionals depend on the gradient of the state. We treat the case of elliptic and non-elliptic second order state laws only in the two-dimensional situation. Our results are based on deep facts about gradient Young measures.
We examine how the use of typical techniques from non-convex vector variational problems can help in understanding optimal design problems in conductivity. After describing the main ideas of the underlying analysis and providing some standard material in an attempt to make the exposition self-contained, we show how those ideas apply to a typical optimal desing problem with two different conducting materials. Then we examine the equivalent relaxed formulation to end up with a new problem whose numerical...
A way of geometrically representing symmetric 2 × 2-gradients is proposed, and a general theorem characterizing sets of gradients is proved. We believe this perspective may help in understanding the structure of gradients and visualizing it. Several non-trivial examples are discussed.
We explicitly introduce and exploit div-curl Young measures to examine optimal design problems governed by a linear state law in divergence form. The cost is allowed to depend explicitly on the gradient of the state. By means of this family of measures, we can formulate a suitable relaxed version of the problem, and, in a subsequent step, put it in a similar form as the original optimal design problem with an appropriate set of designs and generalized state law. Many of the issues involved has been...
An optimal control problem when controls act on the
boundary can also be understood as a variational principle under differential
constraints and no restrictions on boundary and/or initial values. From this
perspective, some existence theorems can be proved when cost functionals
depend on the gradient of the state. We treat the case of elliptic and
non-elliptic second order state laws only in the two-dimensional
situation. Our results are based on deep facts about
gradient Young measures.
We examine how the use of typical techniques from non-convex vector variational problems can help in understanding optimal design problems in conductivity. After describing the main ideas of the underlying analysis and providing some standard material in an attempt to make the exposition self-contained, we show how those ideas apply to a typical optimal desing problem with two different conducting materials. Then we examine the equivalent relaxed formulation to end up with a new problem whose numerical...
In the framework of the linear fracture theory, a commonly-used tool
to describe the smooth evolution of a crack embedded in a bounded domain Ω is the so-called
energy release rate defined as the variation of the mechanical
energy with respect to the crack dimension. Precisely, the
well-known Griffith's criterion postulates the evolution of the
crack if this rate reaches a critical value. In this work, in the anti-plane scalar case, we
consider the shape design problem which consists in optimizing...
An alternative approach for the analysis and the numerical
approximation of ODEs, using a variational framework, is
presented. It is based on the natural and elementary idea of minimizing
the residual of the differential equation measured in a usual norm.
Typical existence results for Cauchy problems can thus be
recovered, and finer sets of assumptions for existence are made explicit. We treat, in particular, the cases of an explicit ODE and a differential inclusion. This approach also allows...
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