  In this paper, we solve the time inconsistent portfolio selection problem by
using different utility functions with a moving target as our constraint. We
solve this problem by finding an equilibrium control under the given definition
as our optimal control. We firstly derive a sufficient equilibrium condition
for second-order continuously differentiable utility funtions. Then we use
power functions of order two, three and four in our problem and find the
respective condtions for obtaining an equilibrium for our different problems.
In the last part of the paper, we consider using another definition of
equilibrium to solve our problem when the utility function that we use in our
problem is the negative part of x and also find the condtions for obtaining an
equilibrium.
