Distributed Continuous-time Resource Allocation with Time-varying Resources under Quadratic Cost Functions

We developed distributed continuous-time algorithms to solve the resource allocation problem with quadratic cost functions and continuously time-varying resources. Since the resources are time-varying, the optimal solution is changing over time. The allocation decision variable should not only find but also track the optimal solution trajectory. In a distributed manner, the agents work collaboratively to find as well as track the optimal solution using local information. Without the local allocation feasibility constraints, a distributed algorithm is designed based on sign function and consensus protocols. The tracking error is proven to vanish in finite time. When the local allocation feasibility constraints are considered, a distributed algorithm based on singular perturbation theory and penalty function is developed. The tracking error is proven to be uniformly ultimately bounded.