A gradient method for solving an optimal control problem described by a parabolic equation is considered. The gradient projection method is applied to solve the problem. The convergence of the projection algorithm is investigated.
In this paper we give an alternative proof of our recent result that totally unrectifiable 1-sets which satisfy a measure-theoretic flatness condition at almost every point and sufficiently small scales, satisfy Besicovitch's 1/2-Conjecture which states that the lower spherical density for totally unrectifiable 1-sets should be bounded above by 1/2 at almost every point. This is in contrast to rectifiable 1-sets which actually possess a density equal to unity at almost every point. Our present method...
Ever since the discovery of the connection between the Menger-Melnikov curvature and the Cauchy kernel in the L norm, and its impressive utility in the analytic capacity problem, higher dimensional analogues have been coveted. The lesson from 1-sets was that any such (nontrivial, nonnegative) expression, using the Riesz kernels for m-sets in R, even in any L norm (k ∈ N), would probably carry nontrivial information on whether the boundedness of these kernels in the appropriate norm implies rectifiability...
An optimal control problem governed by a quasilinear parabolic equation with additional constraints is investigated. The optimal control problem is converted to an optimization problem which is solved using a penalty function technique. The existence and uniqueness theorems are investigated. The derivation of formulae for the gradient of the modified function is explainedby solving the adjoint problem.
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