  Several Earth system components are at a high risk of undergoing rapid and
irreversible qualitative changes or `tipping', due to increasing climate
warming. Potential tipping elements include Arctic sea-ice, Atlantic meridional
overturning circulation, and tropical coral reefs. Amidst such immediate
concerns, it has become necessary to investigate the feasibility of arresting
or even reversing the crossing of tipping thresholds using feedback control. In
this paper, we study the control of an idealized diffusive energy balance model
(EBM) for the Earth's climate; this model has two tipping points due to strong
co-albedo feedback. One of these tipping points is a `small icecap' instability
responsible for a rapid transition to an ice-free climate state under
increasing greenhouse gas (GHG) forcing. We develop an optimal control strategy
for the EBM under different climate forcing scenarios with the goal of
reversing sea ice loss while minimizing costs. We find that effective control
is achievable for such a system, but the cost of reversing sea-ice loss nearly
quadruples for an initial state that has just tipped as compared to a state
before reaching the tipping point. We also show that thermal inertia may delay
tipping leading to an overshoot of the critical GHG forcing threshold. This may
offer a short intervention window (overshoot window) during which the control
required to reverse sea-ice loss only scales linearly with intervention time.
While systems with larger system inertia may have longer overshoot windows,
this increased elbow room comes with a steeper rise in the requisite control
once the intervention is delayed past this window. Additionally, we find that
the requisite control to restore sea-ice is localized in the polar region.
