The usual way a thermostat works is the user specifies goal temperatures in time ranges and the thermostat tries to achieve them. E.g. our thermostat allows, II

The usual way a thermostat works is the user specifies goal temperatures in time ranges and the thermostat tries to achieve them. E.g. our thermostat allows, IIRC, 4 different segments per day so we have morning (warmish), daytime (minimal heating), evening (warmish), night (minimal). The optimization problem them becomes one of achieving these goals. To account for the dynamics / circumstances you can augment the input the thermostat receives from just the current temperature to also include information about the rate of change of temperature (velocity and acceleration). This is the idea behind PID controllers and it allows things like easing off the heating if the house is warming quickly so the thermostat doesn't overshoot the goal.