Problem 1
Find the optimal paths of the control, state, and costate variables that will $$\begin{array}{ll} \text { Maximize } & \int_{0}^{1}\left(y-u^{2}\right) d t \\ \text { subject to } & y^{\prime}=u \\ \text { and } & y(0)=2 \quad y(1) \text { free } \end{array}$$
Problem 2
Solve the following exhaustible resource problem for the optimal extraction path: $$\begin{array}{ll}\text { Maximize } & \int_{0}^{T} \ln (q) e^{-s t} d t \\\\\text { subject to } & s^{\prime}=-q \\ \text { and } & s(0)=s_{0} \quad s(t) \geq 0\end{array}$$
Problem 2
Find the optimal paths of the control, state, and costate variables that will $$\begin{array}{ll} \text { Maximize } & \int_{0}^{8} 6 y d t \\ \text { subject to } & y^{\prime}=y+u \\ & y(0)=10 \quad y(8) \text { free } \\ \text { and } & u(t) \in[0,2] \end{array}$$
Problem 3
Find the optimal paths of the control, state, and costate variables that will $$\begin{array}{ll} \text { Maximize } & \int_{0}^{T}-\left(a u+b u^{2}\right) d t \\ \text { subject to } & y^{\prime}=y-u \\ \text { and } & y(0)=y_{0} \quad y(t) \text { free } \end{array}$$
Problem 4
Find the optimal paths of the control, state, and costate variables that will $$\begin{array}{ll} \text { Maximize } & \int_{0}^{T}\left(y u-u^{2}-y^{2}\right) d t \\ \text { subject to } & y^{\prime}=u \\ \text { and } & y(0)=y_{0} \quad y(t) \text { free } \end{array}$$
Problem 6
Find the optimal paths of the control, state, and costate variables that will \(\begin{array}{ll}\text { Maximize } & \int_{0}^{4} 3 y d t \\ \text { subject to } & y^{\prime}=y+u \\ & y(0)=5 \quad y(4) \geq 300 \\ \text { and } & 0 \leq u(t) \leq 2\end{array}\)
Problem 7
Find the optimal paths of the control, state, and costate variables that will \(\begin{array}{ll}\text { Maximize } & \int_{0}^{1}-u^{2} d t \\ \text { subject to } & y^{\prime}=y+u \\ \text { and } & y(0)=1 \quad y(1)=0\end{array}\)
Problem 9
Find the optimal paths of the control, state, and costate variables that will Maximize \(\int_{0}^{2}\left(2 y-3 u-a u^{2}\right) d t\) subject to \(\quad y^{\prime}=u+y\) and \(\quad y(0)=5 \quad y(2)\) free
