Optimal control

The theory of optimal control ( engl. optimal control theory ) is closely related to the calculus of variations and optimization. Optimum control is a function that minimizes or maximizes a given objective function with a differential equation constraints and possibly other constraints.

For example, a driver may attempt to reach a target in less time as possible. When the driver turns the most? May have certain constraints, such as speed limits are adhered to. Another motorist, however, may be tempted to minimize fuel consumption, that is, he chooses a different objective function.

The optimal control problem

There are several mathematical formulations of the task, in which we specify a general form as possible here.

Let and.

Wanted is a state and a controller, so that:

Subject to the constraints:

A which satisfies this equation is referred to as optimal control.

Also occur frequently still called state constraints, ie The state at a given time is also subject to certain restrictions.

Of interest are primarily the following questions:

While the calculus of variations admissible functions only allowed to open sets, more general conditions (including closed sets for the control functions u) were in the optimal control observed with another formalism that differs between control functions u ( t) and state functions x ( t). The Pontrjaginsche maximum principle is a generalization of the Weierstrass condition of the calculus of variations. For the maximum principle new proof methods (including separation of cones, needle variations) were required.

Economic applications

The methodology of optimal control was applied early on practical areas of the economy. Robert Dorfman put 1969 in front of an economic interpretation of the theory of optimal control.


A company wants to maximize profits over a certain period of time. At any given time, it has a capital stock due to past behavior. If this stock of capital the company can make a decision (for example, in terms of output, price, etc.). If the company and receives a profit per unit of time. It can then formulate a dynamic optimization problem for a time interval: