Apparatus, systems, methods, and related computer program products for carrying out a demand response (DR) event via an intelligent, network-connected thermostat associated with a structure. The systems disclosed include an energy management system in operation with an intelligent, network-connected thermostat located at a structure. The thermostat is operable to control an HVAC system. Control during a DR event period may be performed based on an optimal control trajectory of the HVAC system, where the control trajectory is optimal in that it minimizes a cost function comprising a combination of a first factor representative of a total energy consumption during the DR event period, a second factor representative of a metric of occupant discomfort, and a third factor representative of deviations of a rate of energy consumption over the DR event period.