Hamilton-Jacobi-Bellman equation

The Hamilton--Jacobi--Bellman equation is the continuous-time Bellman equation. For state dynamics $\dot{x}=f(x,u,t)$, running cost $L(x,u,t)$ and terminal cost $\Phi (x)$, a smooth value function satisfies

For a nonnegative classical solution $V$ of a finite-horizon Hamilton-Jacobi-Bellman equation with nonnegative running reward $r$, the process $V(X_t,t)+{\int }_0^{t}r(X_s,a_s)ds$ is a local supermartingale under every admissible control. The Itô formula gives nonpositive drift because the chosen control cannot exceed the supremum in the equation. Stop where the stochastic integral becomes a true martingale, take expectations and apply the Fatou lemma. Nonnegativity prevents loss of a lower bound when removing the stops, giving the verification upper bound. A maximizing control attains the bound when its stopped reward processes are uniformly integrable.

A linear-quadratic optimal-control problem has linear state dynamics and running and terminal costs defined by quadratic forms. Its value function is quadratic in the state, the optimal feedback is linear, and its coefficient obeys a Riccati equation.

A backward scalar or matrix recurrence for the quadratic coefficients in a value function for a linear-quadratic optimal control problem. It results from completing the control square in the Bellman equation. Multiplicative noise modifies the quadratic coefficients through its second moments.