Costate equation

Definition

Let $U \subseteq \R^m$ be nonempty, $T > 0$, and let $f \colon [0, T] \times \R^n \times U \to \R^n$ and $L \colon [0, T] \times \R^n \times U \to \R$ be continuous with continuous partial derivatives in $x$. Let $H(t, x, u, p, \lambda_0) = p \cdot f(t, x, u) - \lambda_0 L(t, x, u)$ be the Hamiltonian, and fix $\lambda_0 \ge 0$. Let $u \colon [0, T] \to U$ be a measurable bounded control with trajectory $x$ of the control system $x' = f(t, x, u)$. The costate equation along $(x, u)$ is the linear differential equation $$p'(t) = -\frac{\partial H}{\partial x}(t, x(t), u(t), p(t), \lambda_0) = -\Big(\frac{\partial f}{\partial x}(t, x(t), u(t))\Big)^{T} p(t) + \lambda_0 \nabla_x L(t, x(t), u(t))$$ for an absolutely continuous function $p \colon [0, T] \to \R^n$, required to hold for almost every $t \in [0, T]$. A solution $p$ is a costate or adjoint vector along $(x, u)$.