Inventory Control General method

A generalization of the calculus of variations, is a mathematical optimization method for deriving control policies. The method is largely due to the work of Lev Pontryagin and his collaborators, summarized in English in Pontryagin (1962).

General method

Optimal control deals with the problem of finding a control law for a given system such that a certain optimality criterion is achieved. A control problem includes a cost functional that is a function of state and control variables. An optimal control is a set of differential equations describing the paths of the control variables that minimize the cost functional. The optimal control can be derived using Pontryagin's maximum principle (a necessary condition), or by solving the Hamilton-Jacobi-Bellman equation (a sufficient condition).

We begin with a simple example. Consider a car traveling on a straight line through a hilly road. The question is, how should the driver press the accelerator pedal in order to minimize the total traveling time? Clearly in this example, the term control law refers specifically to the way in which the driver presses the accelerator and shifts the gears. The "system" consists of both the car and the road, and the optimality criterion is the minimization of the total traveling time. Control problems usually include ancillary constraints. For example the amount of available fuel might be limited, the accelerator pedal cannot be pushed through the floor of the car, speed limits, etc.

A proper cost functional is a mathematical expression giving the traveling time as a function of the speed, geometrical considerations, and initial conditions of the system. It is often the case that the constraints are interchangeable with the cost functional.

Another optimal control problem is to find the way to drive the car so as to minimize its fuel consumption, given that it must complete a given course in a time not exceeding some amount. Yet another control problem is to minimize the total monetary cost of completing the trip, given assumed monetary prices for time and fuel.

A more abstract framework goes as follows. Given a dynamical system with time-varying input u(t), time-varying output y(t) and time-varying state x(t), define a cost functional to be minimized. The cost functional is the sum of the path costs, which usually take the form of an integral over time, and the terminal costs, which is a function only of the terminal (i.e., final) state, x(T). Thus, this cost functional typically takes the form

where T is the terminal time of the system. It is common, but not required, to have the initial (i.e., starting) time of the system be 0 as shown. The minimization of a functional of this nature is related to the minimization of action in Lagrangian mechanics, in which case L(x,u,t) is called the Lagrangian