#####    cont5.mod     ####    modelfile for problem 5   ####
#
param n default 59;
param n1 := n+1;
param n4 := n*4;
param a default .01;

var x{0..n1, 0..n1};
var u{1..n4};

param h := 1/n1;
param h2 := h^2;

param yd{i in 1..n, j in 1..n} := 2 - 2*(i*h*(i*h-1)+j*h*(j*h-1));

minimize f:	.5*h2*sum{i in 1..n, j in 1..n} (x[i,j]-yd[i,j])^2 +
		.5*h*a*sum{i in 1..n4} u[i]^2;

s.t. pde{i in 1..n, j in 1..n}:
	4*x[i,j] - sum{k in {-1,1}} (x[i+k,j] + x[i,j+k]) = 0;

s.t. bc1 {i in 1..n}: x[i,0] - x[i,1] = h * (u[i] - x[i,0]^2);
s.t. bc2 {i in 1..n}: x[n1,i] - x[n,i] = h * (u[n+i] - x[n1,i]^2) ;
s.t. bc3 {i in 1..n}: x[i,n1] - x[i,n] = h * (u[2*n+i] - x[i,n1]^2);
s.t. bc4 {i in 1..n}: x[0,i] - x[1,i] = h * (u[3*n+i] - x[0,i]^2) ;

s.t. sc{i in 0..n1, j in 0..n1}: 0 <= x[i,j] <=  2.071;

s.t. cc{i in 1..n4}: 3.7 <= u[i] <= 4.5;    # control constraints