The value function of an
optimization problem
In mathematics, engineering, computer science and economics
Economics () is a behavioral science that studies the Production (economics), production, distribution (economics), distribution, and Consumption (economics), consumption of goo ...
gives the
value attained by the
objective function
In mathematical optimization and decision theory, a loss function or cost function (sometimes also called an error function) is a function that maps an event or values of one or more variables onto a real number intuitively representing some "cost ...
at a solution, while only depending on the
parameter
A parameter (), generally, is any characteristic that can help in defining or classifying a particular system (meaning an event, project, object, situation, etc.). That is, a parameter is an element of a system that is useful, or critical, when ...
s of the problem. In a
controlled dynamical system
In mathematics, a dynamical system is a system in which a Function (mathematics), function describes the time dependence of a Point (geometry), point in an ambient space, such as in a parametric curve. Examples include the mathematical models ...
, the value function represents the optimal payoff of the system over the interval
1">, t1/var> when started at the time-t state variable x(t)=x. If the objective function represents some cost that is to be minimized, the value function can be interpreted as the cost to finish the optimal program, and is thus referred to as "cost-to-go function." In an economic context, where the objective function usually represents utility
In economics, utility is a measure of a certain person's satisfaction from a certain state of the world. Over time, the term has been used with at least two meanings.
* In a normative context, utility refers to a goal or objective that we wish ...
, the value function is conceptually equivalent to the indirect utility function.
In a problem of optimal control, the value function is defined as the supremum
In mathematics, the infimum (abbreviated inf; : infima) of a subset S of a partially ordered set P is the greatest element in P that is less than or equal to each element of S, if such an element exists. If the infimum of S exists, it is unique, ...
of the objective function taken over the set of admissible controls. Given , a typical optimal control problem is to
:
subject to
:
with initial state variable . The objective function is to be maximized over all admissible controls