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# 数学代写|优化理论代写Optimization Theory代考|PONTRYAGIN’S MINIMUM PRINCIPLE AND STATE INEQUALITY CONSTRAINTS

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## 数学代写|优化理论代写Optimization Theory代考|PONTRYAGIN’S MINIMUM PRINCIPLE AND STATE INEQUALITY CONSTRAINTS

So far, we have assumed that the admissible controls and states are not constrained by any boundaries; however, in realistic systems such constraints do commonly occur. Physically realizable controls generally have magnitude limitations: the thrust of a rocket engine cannot exceed a certain value; motors, which provide torque, saturate; attitude control mass expulsion systems are capable of providing a limited torque. State constraints often arise because of safety, or structural restrictions: the current in an electric motor cannot exceed a certain value without damaging the windings; the turning radius of a maneuvering aircraft cannot be less than a specified minimum value; a spacecraft reentering the earth’s atmosphere must satisfy certain attitude and velocity constraints to avoid burning up.
Let us first consider the effect of control constraints on the fundamental theorem derived in Section 4.1, and then show how the nećessary conditions are modified. $\dagger$ This generalization of the fundamental theorem leads to Pontryagin’s minimum principle.
Pontryagin’s Minimum Principle
By definition, the control $\mathbf{u}^$ causes the functional $J$ to have a relative minimum if $$J(\mathbf{u})-J\left(\mathbf{u}^\right)=\Delta J \geq 0$$
for all admissible controls sufficiently close to $\mathbf{u}^$. If we let $\mathbf{u}=\mathbf{u}^+\delta \mathbf{u}$, the increment in $J$ can be expressed as
$$\Delta J\left(\mathbf{u}^, \delta \mathbf{u}\right)=\delta J\left(\mathbf{u}^, \delta \mathbf{u}\right)+\text { higher-order terms }$$
$\delta J$ is linear in $\delta \mathbf{u}$ and the higher-order terms approach zero as the norm of $\delta$ approaches zero. If we were to re-prove the fundamental theorem for unbounded controls using control system notation, the reasoning would be exactly as given in Section 4.1. That is, if the control were unbounded, we could use the linearity of $\delta J$ with respect to $\delta \mathbf{u}$, and the fact that $\delta \mathbf{u}$ can vary arbitrarily to show that a necessary condition for $\mathbf{u}^$ to be an extremal control is that the variation $\delta J\left(\mathbf{u}^, \delta \mathbf{u}\right)$ must be zero for all admissible $\delta \mathbf{u}$ having a sufficiently small norm. Since we are no longer assuming that the admissible controls are not bounded, $\delta \mathbf{u}$ is arbitrary only if the extremal control is strictly within the boundary for all time in the interval $\left[t_0, t_f\right]$. In this case, the boundary has no effect on the problem solution. If, however, an extremal control lies on a boundary during at least one subinterval $\left[t_1, t_2\right]$ of the interval $\left[t_0, t_f\right]$, as shown in Fig. 5-13(a), then admissible control variations $\delta \hat{\mathbf{u}}$ exist whose negatives $(-\delta \hat{u})$ are not admissible. One such control variation is shown in Fig. 5-13(b). If only these variations are considered, a necessary condition for $\mathbf{u}^$ to minimize $J$ is that $\delta J\left(\mathbf{u}^, \delta \hat{\mathbf{u}}\right) \geq 0$. On the other hand, for variations
$\delta \tilde{u}$, which are nonzero only for $t$ not in the interval $\left[t_1, t_2\right]$, as, for example, in Fig. $5-13(\mathrm{c})$, it is necessary that $\delta J\left(\mathbf{u}^, \delta \tilde{u}\right)=0$; the reasoning used in proving the fundamental theorem applies. Considering all admissible variations with $|\delta \mathbf{u}|$ small enough so that the sign of $\Delta J$ is determined by $\delta J$, we see that a necessary condition for $\mathbf{u}^$ to minimize $J$ is
$$\delta J\left(\mathbf{u}^*, \delta \mathbf{u}\right) \geq 0$$

## 数学代写|优化理论代写Optimization Theory代考|Additional Necessary Conditions

Pontryagin and his co-workers have also derived other necessary conditions for optimality that we will find useful. We now state, without proof, two of these necessary conditions:
If the final time is fixed and the Hamiltonian does not depend explicitly on time, then the Hamiltonian must be a constant when evaluated on an extremal trajectory; that is,
$$\mathscr{H}\left(\mathbf{x}^(t), \mathbf{u}^(t), \mathbf{p}^*(t)\right)=c_1 \quad \text { for } t \in\left[t_0, t_f\right]$$
If the final time is free, and the Hamiltonian does not explicitly depend on time, then the Hamiltonian must be identically zero when evaluated on an extremal trajectory; that is,
$$\mathscr{H}\left(\mathbf{x}^(t), \mathbf{u}^(t), \mathbf{p}^*(t)\right)=0 \quad \text { for } t \in\left[t_0, t_f\right]$$
State Variable Inequality Constraints
Let us now consider problems in which there may be inequality constraints that involve the state variables as well as the controls. It will be assumed that the state constraints are of the form
$$\mathbf{f}(\mathbf{x}(t), t) \geq \mathbf{0 , \dagger}$$
where $\mathbf{f}$ is an $l$-vector function $(l \leq m)$ of the states and possibly time, which has continuous first and second partial derivatives with respect to $\mathbf{x}(t)$. It will also be assumed that the admissible control values lie in a closed and bounded region. Our approach will be to transform the $l$ inequality constraints of (5.3-42) into a single equality constraint, and then to augment the performance measure with this equality constraint, as we have done previously with the state equations.

## 数学代写|优化理论代写Optimization Theory代考|PONTRYAGIN’S MINIMUM PRINCIPLE AND STATE INEQUALITY CONSTRAINTS

Pontryagin最小原则

$$\Delta J\left(\mathbf{u}^, \delta \mathbf{u}\right)=\delta J\left(\mathbf{u}^, \delta \mathbf{u}\right)+\text { higher-order terms }$$
$\delta J$在$\delta \mathbf{u}$中是线性的，并且随着$\delta$的范数趋于零，高阶项趋于零。如果我们要用控制系统符号重新证明无界控制的基本定理，其推理将完全如第4.1节所述。也就是说，如果控制是无界的，我们可以使用$\delta J$相对于$\delta \mathbf{u}$的线性，并且$\delta \mathbf{u}$可以任意变化的事实表明，$\mathbf{u}^$成为极端控制的必要条件是，对于所有允许的$\delta \mathbf{u}$具有足够小的范数，变化$\delta J\left(\mathbf{u}^, \delta \mathbf{u}\right)$必须为零。因为我们不再假设允许的控制是无界的，所以$\delta \mathbf{u}$是任意的，只有当极限控制严格地在区间$\left[t_0, t_f\right]$的所有时间内的边界内。在这种情况下，边界对问题的解决没有影响。然而，如果在区间$\left[t_0, t_f\right]$的至少一个子区间$\left[t_1, t_2\right]$的边界上有一个极值控制，如图5-13(a)所示，则存在可容许的控制变异$\delta \hat{\mathbf{u}}$，其负$(-\delta \hat{u})$是不可容许的。图5-13(b)显示了一种这样的控制变化。如果只考虑这些变化，那么$\mathbf{u}^$使$J$最小化的一个必要条件是$\delta J\left(\mathbf{u}^, \delta \hat{\mathbf{u}}\right) \geq 0$。另一方面，对于变化
$\delta \tilde{u}$，它只在$t$不在$\left[t_1, t_2\right]$区间内是非零的，例如图$5-13(\mathrm{c})$，则需要$\delta J\left(\mathbf{u}^, \delta \tilde{u}\right)=0$;用来证明基本定理的推理是适用的。考虑到$|\delta \mathbf{u}|$的所有允许的变化都足够小，以至于$\Delta J$的符号由$\delta J$决定，我们看到$\mathbf{u}^$使$J$最小的必要条件是
$$\delta J\left(\mathbf{u}^*, \delta \mathbf{u}\right) \geq 0$$

## 数学代写|优化理论代写Optimization Theory代考|Additional Necessary Conditions

$$\mathscr{H}\left(\mathbf{x}^(t), \mathbf{u}^(t), \mathbf{p}^(t)\right)=c_1 \quad \text { for } t \in\left[t_0, t_f\right]$$ 如果最终时间是自由的，并且哈密顿量不显式地依赖于时间，那么在极值轨迹上计算哈密顿量时，哈密顿量必须等于零;也就是说， $$\mathscr{H}\left(\mathbf{x}^(t), \mathbf{u}^(t), \mathbf{p}^(t)\right)=0 \quad \text { for } t \in\left[t_0, t_f\right]$$

$$\mathbf{f}(\mathbf{x}(t), t) \geq \mathbf{0 , \dagger}$$

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