Динамикалық жүйелерді басқарудың оптималды теориясы

Mathematical way of attaining a desired output from a dynamic system

Optimal control theory is a branch of control theory that deals with finding a control for a dynamical system over a period of time such that an objective function is optimized. It has numerous applications in science, engineering and operations research. For example, the dynamical system might be a spacecraft with controls corresponding to rocket thrusters, and the objective might be to reach the Moon with minimum fuel expenditure. Or the dynamical system could be a nation's economy, with the objective to minimize unemployment; the controls in this case could be fiscal and monetary policy. A dynamical system may also be introduced to embed operations research problems within the framework of optimal control theory. Optimal control is an extension of the calculus of variations, and is a mathematical optimization method for deriving control policies. The method is largely due to the work of Lev Pontryagin and Richard Bellman in the 1950s, after contributions to calculus of variations by Edward J. McShane. Optimal control can be seen as a control strategy in control theory. or by solving the Hamilton–Jacobi–Bellman equation (a sufficient condition). We begin with a simple example. Consider a car traveling in a straight line on a hilly road. The question is, how should the driver press the accelerator pedal in order to minimize the total traveling time? 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 function will be a mathematical expression giving the traveling time as a function of the speed, geometrical considerations, and initial conditions of the system. Constraints are often interchangeable with the cost function. Another related optimal control problem may be 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 related control problem may be 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.

The examples thus far have shown continuous time systems and control solutions. In fact, as optimal control solutions are now often implemented digitally, contemporary control theory is now primarily concerned with discrete time systems and solutions. The Theory of Consistent Approximations provides conditions under which solutions to a series of increasingly accurate discretized optimal control problem converge to the solution of the original, continuous time problem. Not all discretization methods have this property, even seemingly obvious ones. For instance, using a variable step size routine to integrate the problem's dynamic equations may generate a gradient which does not converge to zero (or point in the right direction) as the solution is approached. The direct method RIOTS is based on the Theory of Consistent Approximation.

A common solution strategy in many optimal control problems is to solve for the costate (sometimes called the shadow price) The costate summarizes in one number the marginal value of expanding or contracting the state variable next turn. The marginal value is not only the gains accruing to it next turn but associated with the duration of the program. It is nice when can be solved analytically, but usually, the most one can do is describe it sufficiently well that the intuition can grasp the character of the solution and an equation solver can solve numerically for the values. Having obtained , the turn t optimal value for the control can usually be solved as a differential equation conditional on knowledge of Again it is infrequent, especially in continuous time problems, that one obtains the value of the control or the state explicitly. Usually, the strategy is to solve for thresholds and regions that characterize the optimal control and use a numerical solver to isolate the actual choice values in time.

Consider the problem of a mine owner who must decide at what rate to extract ore from their mine. They own rights to the ore from date to date At date there is ore in the ground, and the time dependent amount of ore left in the ground declines at the rate of that the mine owner extracts it. The mine owner extracts ore at cost (the cost of extraction increasing with the square of the extraction speed and the inverse of the amount of ore left) and sells ore at a constant price Any ore left in the ground at time cannot be sold and has no value (there is no "scrap value"). The owner chooses the rate of extraction varying with time to maximize profits over the period of ownership with no time discounting.