Using NDSolve on wave PDE on string, when solution given at 2 different times instead of initial velocity?

I don't have time to make a solid answer, but this seems to work :

pde = -D[u[x, t], {t, 2}] + u[x, t] == 
   D[u[x, t], {x, 2}] + 2*Exp[-t]*(x - (1/2)*x^2 + (1/2)*t - 1) + 
    NeumannValue[0, x == 1];

bc = {u[0, t] == 0};

ic = {u[x, 0] == x^2 - 2*x
   , PeriodicBoundaryCondition[
    u[x, t] - (((1/2)*x^2 - x)*Exp[-1] - ((3*x^2)/4 - (3/2)*x)*
        Exp[-2^(-1)])
    , t == 1 && 0 < x < 1
    , Function[xy, xy - {0, 1/2}]]};

U = NDSolveValue[{pde, ic, bc}, u, {x, 0, 1}, {t, 0, 1}];
Plot3D[U[x, t], {x, 0, 1}, {t, 0, 1}, AxesLabel -> {x, t, u}]  

Plot[{
  U[x, 0]
  , U[x, 1/2]
  , U[x, 1]
  , U[x, 1/2] + ((1/2)*x^2 - x)*Exp[-1] - ((3*x^2)/4 - (3/2)*x)*
    Exp[-2^(-1)]}, {x, 0, 1}, 
 PlotStyle -> {Red, Green, Directive[Blue, AbsoluteThickness[7]], 
   Directive[Black, Dashed, AbsoluteThickness[3]]}, 
 PlotLegends -> "Expressions"]  

Here is the error :

Plot3D[Evaluate[-D[U[x, t], {t, 2}] + 
   U[x, t] - (D[U[x, t], {x, 2}] + 
     2*Exp[-t]*(x - (1/2)*x^2 + (1/2)*t - 1))], {t, 0, 1}, {x, 0, 1}]  

The method automaticaly chosen by NDSolve is Method -> {"PDEDiscretization" -> {"FiniteElement"}} (as opposed to Method] -> {"PDEDiscretization" -> {"MethodOfLines", "SpatialDiscretization" -> {"FiniteElement", femopts}}}) . This is the reason why one can impose boundaries condition on the variable "time".

Note also that the term "PeriodicBoundaryCondition" is a little bit misleading because the source of the "boundary condition" does not need to be a boundary.