  In this article, we present two different approaches for obtaining
quantitative inequalities in the context of parabolic optimal control problems.
Our model consists of a linearly controlled heat equation with Dirichlet
boundary condition $(u_f)_t-\Delta u_f=f$, $f$ being the control. We seek to
maximise the functional $\mathcal J_T(f):=\frac12\int_{(0;T)\times \Omega}
u_f^2$ or, for some $\epsilon>0$, $\mathcal J_T^\epsilon
(f):=\frac12\int_{(0;T)\times \Omega} u_f^2+\epsilon \int_\Omega
u_f^2(T,\cdot)$ and to obtain quantitative estimates for these maximisation
problems. We offer two approaches in the case where the domain $\Omega$ is a
ball. In that case, if $f$ satisfies $L^1$ and $L^\infty$ constraints and does
not depend on time, we propose a shape derivative approach that shows that, for
any competitor $f=f(x)$ satisfying the same constraints, we have $\mathcal
J_T(f^*)-\mathcal J_T(f)\gtrsim \Vert f-f^*\Vert_{L^1(\Omega)}^2$, $f^*$ being
the maximiser. Through our proof of this time-independent case, we also show
how to obtain coercivity norms for shape hessians in such parabolic
optimisation problems. We also consider the case where $f=f(t,x)$ satisfies a
global $L^\infty$ constraint and, for every $t\in (0;T)$, an $L^1$ constraint.
In this case, assuming $\epsilon>0$, we prove an estimate of the form $\mathcal
J_T^\epsilon (f^*)-\mathcal J_T^\epsilon (f)\gtrsim\int_0^T a_\epsilon (t)
\Vert f(t,\cdot)-f^*(t,\cdot)\Vert_{L^1(\Omega)}^2$ where $a_\epsilon (t)>0$
for any $t\in (0;T)$. The proof of this result relies on a uniform bathtub
principle.
