State equations partly time varying
"Controllability" refers to being able to change states, to any arbitrary target, by means of the inputs. If A and B are time-invariant, then inputs-to-states relation is always the same. Therefore, the controllability matrix rank can tell you if the system is controllable.

To say that C changes with time is to say that the states-to-outputs relation changes. This means the system is not observable if \$t\$ is not known, but if \$t\$ is known, the system is possibly observable. One can evaluate \$C(t)\$, obtain an observability matrix, and check its rank.

The following matlab code shows that your system is neither controllable or observable for \$t\$ from 0 to 100.

A = [-1 0; 0 -1];
B = [1; 1];
rank(ctrb(A,B)) % rank = 1, system is not controlable

t = 0:100;
R = zeros(1,length(t));
for i = 1:length(t)
    C = [t(i)^2 1];
    R(i) = rank(obsv(A,C));
end
find(R == 2) % empty, system is not observable in evaluated time range