In the framework of optimal control we have introduced in class the HJB equation to solve optimal control problem of the form
$$ \inf_{u\in\mathcal{U}}\left[\int_{0}^{T}L(t,x_u(t),u(t))dt + h(x_u(T));\quad x’(t)=f(t,x(t),u(t)), x(0)=x_0\right] $$
Where $x(t)\in\mathbb{R}^{d}$, $u(t)$ is a control, $L$ a cost function which is continuous, $h$ some function that is differentiable, $\mathcal{U}$ a set of admissible control and $f$ a function satisfying the hypotheses of the Cauchy Lipschitz theorem.
The interest of HJB has been described as follows : when the only theorem you know about sufficient condition (in the « Pontryagin setting ») of optimality fails, use the following result :
If the value function $v(t,x(t))$ is a solution $C^1$ of the HJB : $\partial_t v(t,x) + H^{*}(t,x(t),\partial_x v(t,x)) = 0$ and if $\forall (t,x_t)\in[0,T]\times\mathbb{R}^{d}$ there exists $u(t,x(t))$ minimizing the Hamiltonian with respect to the control variable. Then $x(t),u(t,x(t))$ is optimal in the minimization problem.
In practice, when it is solvable (linear quadratic case) we don’t explicitly right the value function but start with a guess function and see if it solved the HJB. Until now I am fine, however often when the guess was right, they say that is the value function because it solved the HJB equation, however I am not aware of the uniqueness of the solution and so that it is the value function.
So my question is the following : do we have uniqueness result for the HJB equation in the framework of optimal control ?
Thank you a lot !
The value function is defined $\forall (t,x_t)$ (where $t\in[0,T]$ and $x_t$ is an initial condition at time $t$) by
$$ v(t,x_t) =\inf_{u\in\mathcal{U}_t}\left[\int_{0}^{T}L(t,x_u(t),u(t))dt + h(x_u(T))\right] $$