In Dynamic Data-Driven Application Systems (DDDAS), applications must dynamically adapt their behavior in response to objectives and conditions that change while deployed. Often these applications may be safety critical or tightly resource constrained, with a need for graceful degradation when introduced to unexpected conditions. This paper begins by motivating and providing a vision for a dynamically adaptable mixed critical computing platform to support DDDAS applications. We then specifically focus on the need for advancements in task models and scheduling algorithms to manage the resources of such a platform. We discuss the short comings of existing task models for capturing important attributes of our envisioned computing platform, and identify challenges that must be addressed when developing scheduling algorithms that act upon our proposed extended task model.