Objective function doesn't converge but jumps around

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I have been trying to complete an optimisation problem. I am looking at radial heat diffusion on a cylinder. I want to tackle it as an inverse problem in order to obtain values for heat diffusion and the like. I have an objective function Boundary_temperature which spits out the difference between the simulated data(computed by Battery_temp) and the measured data(contained in synth_temp).
The value of the objective function is just the l^2 norm of the calculated data and measured data, so a good value of the parameters should yield a small value. I set up the problem and the value of the objective function jumped about and didn't converge but jumped around, I even managed to obtain a warning of the matrix is singular warning which I don't understand.
I added in a stopping criterion, TolFun, which I set to 10^-3, and I hoped that would make things better, as I know that the value of the objective function did get that small, so I hoped that it would work quite nicely, I was wrong.
The equation in question is:
, ,
The variables I am trying to find is .
Matthew Hunt
Matthew Hunt on 2 Mar 2020
I've added in some equations to make things clear.

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