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The value function of an optimization problem gives the value attained by the objective function at a solution, while only depending on the parameters of the problem. In a controlled dynamical system, the value function represents the optimal payoff of the system over the interval when started at the time-t state variable x=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, 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 of the objective function taken over the set of admissible controls. Given ∈ × R d {\displaystyle \in \times \mathbb {R} ^{d}} , a typical optimal control problem is to
subject to
with initial state variable x = x 0 {\displaystyle x=x_{0}}. The objective function J {\displaystyle J} is to be maximized over all admissible controls u ∈ U {\displaystyle u\in U} , where u {\displaystyle u} is a Lebesgue measurable function from {\displaystyle } to some prescribed arbitrary set in R m {\displaystyle \mathbb {R} ^{m}}. The value function is then defined as