Topics considered here include: examples of optimal control problems; dynamic programming and the Hamilton-Jacobi-Bellman equation; verification theorems; the Pontryagin Maximum Principle Principle. The examples include many with an economic flavor, but others too (including the Hopf-Lax solution formula for ut + H(Du) = 0 with H convex). There’s much more here than we’ll have time to do in lec...

This article considers a discrete-time robust optimal control problem on matrix Lie groups. The underlying system is assumed to be perturbed by exogenous unmeasured bounded disturbances, and the posed as min–max wherein disturbance adversary tries maximize cost that minimize. Assuming existence of saddle point in problem, we present version Pontryagin maximum principle (PMP) encapsulates first-...

Optimal control theory is a powerful mathematical tool, which has known rapid development since the 1950s, mainly for engineering applications. More recently, it become widely used method to improve process performance in quantum technologies by means of highly efficient dynamics. This tutorial aims at providing an introduction key concepts optimal that accessible physicists and engineers worki...

In this paper, the min-time optimal control problem is mainly investigated in the linear time invariant (LTI) continuous-time control system with a constrained input. A high order dynamical LTI system is firstly considered for this purpose. Then the Pontryagin principle and some necessary optimality conditions have been simultaneously used to solve the optimal control problem. These optimality ...

A train travels from one station to the next along a level track. The journey must be completed within a given time and it is desirable to minimise the energy required to drive the train. It has been shown with an appropriate formulation of the problem that an optimal strategy exists and that this strategy must satisfy a Pontryagin type criterion. In this paper the Pontryagin principle will be ...

This letter discusses stochastic optimal control of an energy storage system (ESS) for reducing the impact on grid fast charging electric vehicles in a area. A trade off is achieved between objectives limiting power exchanged with grid, and one fluctuation, around given reference, ESS energy. We show that solution problem can be derived from related deterministic problem, requiring realistic as...

We consider the Lagrange problem of optimal control with unrestricted controls and address the question: under what conditions we can assure optimal controls are bounded? This question is related to the one of Lipschitzian regularity of optimal trajectories, and the answer to it is crucial for closing the gap between the conditions arising in the existence theory and necessary optimality condit...