Can MATLAB solve it? Yes. Your doubt is only you not knowing how to solve the problem.
In fact, simplest is to transform the problem. Your goal is to solve the problem:
minimize x^sqrt(y)
subject to the constraints
1.01 <= x <= 100
1.01 <= y <= 2
x^y >= 100
I'll solve the problem by use of a simple constraint. Remember that the log is a monotonic transform. So if minimize something, then I will still minimize it if I take the log, and take the mimimum of that.
My variation of you problem uses the transformation u = log(x). Assume that log here means the natural log, what MATLAB gives you in the log function. First, recognize these two identities
log(x^sqrt(y)) = log(x) * sqrt(y)
and
log(x^y) = log(x)*y
Do you see this works nicely? So my version of the problem is much simpler:
minimize u*sqrt(y)
subject to the constraints
log(1.01) <= u <= log(100)
1.01 <= y <= 2
u*y >= log(100)
I have a funny feeling I can set that up in a symbolic form, solving it with Lagrange multipliers on paper. But we can use a numerical solver as you did:
u = optimvar('u');
y = optimvar('y');
prob = optimproblem( "Objective", u*sqrt(y));
prob.Constraints.con1 = u >= log(1.01);
prob.Constraints.con2 = y >= 1.01;
prob.Constraints.con3 = y <= 2;
prob.Constraints.con4 = u*y >= log(100);
I see at this point that you defined prob.Constraints.con4 TWICE. That surely caused a problem in your solve.
prob.Constraints.con5 = u <= log(100);
V0.u = 2;
V0.y = 1.5;
sol = solve(prob,V0)
x = exp(sol.u)
y = sol.y
SURPRISE! It worked. As I suggested above, your problem was most likely in the definition of con4. Regardless, the use of logs made the problem a bit simpler to solve. The use of powers often creates numerical problems. Things get nasty really fast then. But most important was probably to just write more careful code.
