Optimal Control Solution to Nonlinear Causal Operator Systems with Target State

We will study the optimal control solution satisfying nonlinear causal operator differential equations. The transformation by these operators depends on the information about the trajectory as a function of past history which will not be affected by future events. These dynamical systems involve a large class of delay, integral, composition, or Cartesian products of operators. The optimal solutions for operator dynamical systems with target states will be studied. Title: Computation of the vertical velocity components for a three dimensional convective flow in a mushy layer Abstract: Here we consider a nonlinear three dimensional convective flow in a mushy layer which is cooled from below. During solidification, fluid flow within the mushy layer can cause vertical chimneys or channels void of solid. These chimneys can generate imperfections in the final form of the solidified alloy. A quadratic nonlinear evolution equation satisfied by the amplitude is derived for the hexagonal cells. Then we compute the linear and first order vertical velocity components for super and sub critical cases.