Can I use a solid state relay to control a DC current?
It depends on the relay. Many solid state relays use a thyristor as their main switching component, and a thyristor can't be turned off simply by removing the trigger voltage. They rely on the reverse-bias portion of the AC cycle to switch off. Other solid state relays use FETs as their switching component and those relays may work with DC. It all depends on what the specific relay is rated for, and the only way to know is to carefully read the relay's data sheet.