Discuss what kind of changes to the necessary conditions of optimality in the form of Euler-Lagrange equations must be made if the final time is relaxed. Consider two cases: first, the state at the final time is free, second, the state at the final time must be on some (hyper)surface.
Explain the essence of Pontryagin’s principle of maximum. Give the full first-order necessary conditions of optimality (for optimal control) using Pontryagin’s principle (instead of the equation of stationarity).
Give (a sketch of) the derivation of time-optimal control for a double integrator with bounds on the control signal – bang-bang control. Discuss the key properties of the resulting controller. Discuss also the practical issues related to implementation. What kind of heuristics could be used to compensate those issues?
Skills (use the knowledge to solve a problem)
Design a constrained optimal (possibly minimum-time) controller for a continuous-time systems by invoking Pontryagin’s principle of maximum.