Continuous-Time Lagrange Relaxation for Systems with Delays

Goal: Find a useful heuristic expression for the dual variable in the problem. This exploration will help find initial guesses to optimize other problems.

In this project, we control the flow of water between two water tanks. The water from Tank 1 flows into Tank 2, producing electricity, and the water from Tank 2 exits the system, also generating electricity. The electricity production of each tank depends on the water levels in the tanks and the water flow from them.

The challenge arises from the fact that the water from the first tank takes a positive amount of time to reach the second tank, making the system non-Markovian unless the flowing water is considered as part of the state. Now, if we consider a continuous-time problem where the controls are also continuous, the flow of water over time becomes a bounded function, making the corresponding state infinite-dimensional.

For this reason, we propose relaxing the constraint that connects the tanks and providing an intuitive approach to find the Lagrange multiplier (the optimal dual variable).