The design process (design work) determines the requirements necessary for a process, service or product to fulfill needs. The design process usually results in a specification but may be subtle such as selecting an item from a catalog.
Process design specifies the inputs and actions necessary to eliminate (or render harmless) process errors thereby resulting in delivery of services and products that fulfill requirements.
A design control would, for example, ensure the design inputs are accurate and complete enough for the next iteration of the design process. A design control is meant to avoid wasted effort.