A continuous-time control problem is defined in terms of state variable $x(t) \in \mathbb{R}^{n}$ and control $u(t) \in \mathbb{R}^{m}, 0 \leqslant t \leqslant T$. We desire to minimize $\int_{0}^{T} c(x, t) d t+K(x(T))$, where $T$ is fixed and $x(T)$ is unconstrained. Given $x(0)$ and $\dot{x}=a(x, u)$, describe further boundary conditions that can be used in conjunction with Pontryagin's maximum principle to find $x, u$ and the adjoint variables $\lambda_{1}, \ldots, \lambda_{n}$.

Company 1 wishes to steal customers from Company 2 and maximize the profit it obtains over an interval $[0, T]$. Denoting by $x_{i}(t)$ the number of customers of Company $i$, and by $u(t)$ the advertising effort of Company 1 , this leads to a problem

$$\operatorname{minimize} \int_{0}^{T}\left[x_{2}(t)+3 u(t)\right] d t$$

where $\dot{x}_{1}=u x_{2}, \dot{x}_{2}=-u x_{2}$, and $u(t)$ is constrained to the interval $[0,1]$. Assuming $x_{2}(0)>3 / T$, use Pontryagin's maximum principle to show that the optimal advertising policy is bang-bang, and that there is just one change in advertising effort, at a time $t^{*}$, where

$$3 e^{t^{*}}=x_{2}(0)\left(T-t^{*}\right) .$$