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数学代写|优化理论代写Optimization Theory代考|MINIMUM-TIME PROBLEMS

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数学代写|优化理论代写Optimization Theory代考|MINIMUM-TIME PROBLEMS

In this section we shall consider problems in which the objective is to transfer a system from an arbitrary initial state to a specified target set in minimum time. The target set (which may be moving) will be denoted by
$\dagger$ Performing the differentiation $\partial \mathscr{H} / \partial x_2$ formally also results in the presence of two unit impulse functions, which occur at $x_2^*(t)= \pm 2$; however, these terms are such that either the impulse functions or their coefficients are zero for all $t \in\left[t_0, t_f\right]$, so the impulses do not affect the solution.
$S(t)$, and the minimum time required to reach the target set by $t^$. Mathematically, then, our problem is to transfer a system $$\dot{\mathbf{x}}(t)=\mathbf{a}(\mathbf{x}(t), \mathbf{u}(t), t)$$ from an arbitrary initial state $\mathbf{x}0$ to the target set $S(t)$ and minimize $$J(\mathbf{u})=\int{t_0}^{t_t} d t=t_f-t_0$$
Typically, the control variables may be constrained by requirements such as
$$\left|u_i(t)\right| \leq 1, \quad i=1,2, \ldots, m, \quad t \in\left[t_0, t^\right]$$
Our approach will be to use the minimum principle to determine the optimal control law.†
To introduce several important aspects of minimum-time problems, let us consider the following simplified intercept problem.

数学代写|优化理论代写Optimization Theory代考|The Set of Reachable States

If a system can be transferred from some initial state to a target set by applying admissible control histories, then an optimal control exists and may be found by determining the admissible control that causes the system to reach the target set most quickly. A description of the target set is assumed to be known; thus, to investigate the existence of an optimal control it is useful to introduce the concept of reachable states.

If a system with initial state $\mathbf{x}\left(t_0\right)=\mathbf{x}_0$ is subjected to all admissible control histories for a time interval $\left[t_0, t\right]$, the collection of state values $\mathbf{x}(t)$ is called the set of states that are reachable (from $\mathbf{x}_0$ ) at time $t$, or simply the set of reachable states.

Although the set of reachable states depends on $\mathbf{x}0, t_0$, and on $t$, we shall denote this set by $R(t)$. The following example illustrates the concept of reachable states. Example 5.4-2. Find the set of reachable states for the system $$\dot{x}(t)=u(t)$$ where the admissible controls satisfy $$-1 \leq u(t) \leq 1$$ The solution of Eq. (5.4-9) is $$x(t)=x_0+\int{t_0}^t u(\tau) d \tau$$
If only admissible control values are used, Eq. (5.4-11) implies that
$$x_0-\left[t-t_0\right] \leq x(t) \leq x_0+\left[t-t_0\right] .$$

数学代写|优化理论代写Optimization Theory代考|MINIMUM-TIME PROBLEMS

$\dagger$执行微分$\partial \mathscr{H} / \partial x_2$也会得到两个单位脉冲函数，它们出现在$x_2^*(t)= \pm 2$;然而，这些项使得脉冲函数或它们的系数对所有$t \in\left[t_0, t_f\right]$都为零，所以脉冲不影响解。
$S(t)$，以及达到$t^$设定的目标所需的最短时间。从数学上讲，我们的问题是将系统$$\dot{\mathbf{x}}(t)=\mathbf{a}(\mathbf{x}(t), \mathbf{u}(t), t)$$从任意初始状态$\mathbf{x}0$转移到目标集$S(t)$并最小化$$J(\mathbf{u})=\int{t_0}^{t_t} d t=t_f-t_0$$

$$\left|u_i(t)\right| \leq 1, \quad i=1,2, \ldots, m, \quad t \in\left[t_0, t^\right]$$

数学代写|优化理论代写Optimization Theory代考|The Set of Reachable States

$$x_0-\left[t-t_0\right] \leq x(t) \leq x_0+\left[t-t_0\right] .$$

MATLAB代写

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