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 state variable. 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, a typical optimal control problem is to

subject to

. is a Lebesgue measurable function. The value function is then defined as is called a feedback control policy, or simply a policy function.

as "new" initial condition must be optimal for the remaining problem. If the value function happens to be continuously differentiable, this gives rise to an important partial differential equation known as Hamilton–Jacobi–Bellman equation,

where the maximand on the right-hand side can also be re-written as the Hamiltonian, as

playing the role of the costate variables.,

which after replacing the appropriate terms recovers the costate equation

is Newton notation for the derivative with respect to time.

The value function is the unique viscosity solution to the Hamilton–Jacobi–Bellman equation. In an online closed-loop approximate optimal control, the value function is also a Lyapunov function that establishes global asymptotic stability of the closed-loop system.

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