To Formulation of the general optimal control problem. Sufficient conditions for the existence of an optimal control. The Pontryagin maximum principle: necessary conditions satisfied by an optimal control and sufficiency under additional convexity assumptions.

这是一份Bath巴斯大学MA30061作业代写的成功案

$$
u^{}=-R^{-1} G^{\prime}(P x+b)=K^{\prime} x-R^{-1} G^{\prime} b $$ where $P(\cdot)$ is the solution of $$ -\dot{P}=P F+F^{\prime} P-P G R^{-1} G^{\prime} P+Q \quad P(T)=0 $$ and $b(\cdot)$ is the solution of $$ -\dot{b}=\left(F-G R^{-1} G^{\prime} P\right)^{\prime} b-Q \bar{x} \quad b(T)=0 $$ The minimum index is $$ V^{}\left(x\left(t_{0}\right), t_{0}\right)=x^{\prime}\left(t_{0}\right) P\left(t_{0}\right) x\left(t_{0}\right)+2 x^{\prime}\left(t_{0}\right) b\left(t_{0}\right)+c\left(t_{0}\right)
$$
where $c\left(t_{0}\right)$ is determined from
$$
\dot{c}=b^{\prime} G R^{-1} G^{\prime} b-\bar{x}^{\prime} Q \dot{x} \quad c(T)=0
$$

MA30061 COURSE NOTES ：

shall show that $b(t)$ for all finite $t$ approaches a finite quantity $\bar{b}(t)$. and the fact that $K=-\bar{P} G R^{-1}$ in the limiting case, we obtain, as $T \rightarrow \infty$
$$
\bar{b}(t)=-\int_{t}^{x} \exp \left[\left(F+G K^{r}\right)^{\prime}(\tau-t)\right] Q \dot{x}(\tau) d \tau
$$
Note that differentiation of yields
$$
\dot{b}=-\left(F+G K^{\prime}\right)^{\prime} \bar{b}+Q \tilde{x}
$$
Because $\left(F+G K^{\prime}\right)^{\prime}$ has all eigenvalues with negative real parts, under the stabilizability and detectability assumptions, it is easy to establish that bounded $\tilde{x}(\cdot)$ implies bounded $\bar{b}(\cdot)$ : to see this, suppose that
$$
\left|\exp \left(F+G K^{\prime}\right) t\right| \leq \alpha \exp (-\beta t) \quad \alpha, \beta>0
$$