![]() 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.Ī more abstract framework goes as follows. Constraints are often interchangeable with the cost function.Īnother 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. 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.Ī 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. Control problems usually include ancillary constraints. The system consists of both the car and the road, and the optimality criterion is the minimization of the total traveling time. 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. Consider a car traveling in a straight line on a hilly road. The optimal control can be derived using Pontryagin's maximum principle (a necessary condition also known as Pontryagin's minimum principle or simply Pontryagin's principle), or by solving the Hamilton–Jacobi–Bellman equation (a sufficient condition). An optimal control is a set of differential equations describing the paths of the control variables that minimize the cost function. A control problem includes a cost functional that is a function of state and control variables. Optimal control deals with the problem of finding a control law for a given system such that a certain optimality criterion is achieved. Optimal control can be seen as a control strategy in control theory. 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. Optimal control is an extension of the calculus of variations, and is a mathematical optimization method for deriving control policies. A dynamical system may also be introduced to embed operations research problems within the framework of optimal control theory. 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. 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. It has numerous applications in science, engineering and operations research. 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. Mathematical way of attaining a desired output from a dynamic system Optimal control problem benchmark (Luus) with an integral objective, inequality, and differential constraint
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