Posted on Categories:Ordinary Differential Equations, 常微分方程, 数学代写

# 数学代写|常微分方程代考Ordinary Differential Equations代写|Existence of optimal controls

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## 数学代写|常微分方程代考Ordinary Differential Equations代写|Existence of optimal controls

The first question that arises is if at least one control function exists that solves (10.4). This depends on the components of the problem and on the functional space where the control is sought. For the functional analysis results that we mention in this section, we refer to, e.g., $[17,35]$ and results given in the Appendix.

Now, we discuss a so-called linear-quadratic control problem that allows us to illustrate some classical techniques to prove existence of optimal controls. For this purpose, consider the following linear ODE problem
$$y^{\prime}=A y+B u, \quad y(a)=y_a$$
where $y(x) \in \mathbb{R}^n, u(x) \in \mathbb{R}^m, A \in \mathbb{R}^{n \times n}$, and $B \in \mathbb{R}^{n \times m}$. The solution to this problem is given by
$$y(x)=e^{(x-a) A} y_a+e^{x A} \int_a^x e^{-s A} B u(s) d s .$$
Assuming $u \in L^2\left((a, b) ; \mathbb{R}^m\right)$, and because $\sup _{s \in[a, b]}\left|e^{-s A} B\right|<\infty$, the integrand is also in $L^2$ and the solution $y \in C\left([a, b] ; \mathbb{R}^n\right)$. Thus, (10.7) defines a function $S: L^2\left((a, b) ; \mathbb{R}^m\right) \rightarrow C\left([a, b] ; \mathbb{R}^n\right), u \mapsto y=S(u)$, which is called the control-to-state map. Notice that this map includes a given fixed initial condition.

As it appears in (10.7), this map is affine: $\tilde{S}(u)=S(u)-S(0)$, and $\tilde{S}$ is continuous since
$$|\tilde{S}(u)|_{\infty}=\max {t \in[a, b]}|S(u)(t)-S(0)(t)|_2 \leq c \sqrt{(b-a)}|u|{L^2\left((a, b) ; \mathbb{R}^m\right)}$$
where $|\cdot|_2$ denotes the Euclidean norm.

## 数学代写|常微分方程代考Ordinary Differential Equations代写|Optimality conditions

We have seen that, with a well-defined control-to-state map that encodes the solution of the differential constraint, an optimal control problem becomes a problem of the calculus of variation with the reduced cost functional as follows:
$$\min {u \in U{a d}} \hat{J}(u)$$
Hence, if $\hat{J}$ is differentiable, the optimality conditions for (10.10) can be formulated in terms of functional derivatives. In particular, in terms of the gradient of $\hat{J}$ with respect to the control function, if $u$ is an optimal control, it must satisfy
$$(\nabla \hat{J}(u), v-u) \geq 0, \quad v \in U_{a d} .$$
Now, since $\hat{J}(u)=J(S(u), u)$, differentiability of $\hat{J}$ requires differentiability of $S(\cdot)$ and of $J(\cdot, \cdot)$. Notice that, if $J$ is defined as in (10.4), the conditions $\ell \in C^1$ and $g \in C^1$ are sufficient for guaranteeing the differentiability of $J$.
In the linear-quadratic optimal control case (10.8), the control-to-state map $S$ is an affine function and thus differentiable, and assuming $\ell \in C^1$ and $g \in C^1$, we can state differentiability of $\hat{J}$.
More in general, one defines the map
$$c: H \times U \rightarrow Z, \quad(y, u) \mapsto y^{\prime}-f(\cdot, y, u),$$
where $Z \subseteq U$, such that the differential constraint is formulated as $c(y, u)=0$, where we assume that a given fixed initial condition is included. Then the construction of $S$ requires that the equation $c(y, u)=0$ can be solved for $y$ with a given $u$. Equivalently, this means that $c$ is invertible with respect to $y$. Similarly, if $c$ is differentiable, we have the linearised constraint $\partial c(y, u)(\delta y, \delta u)=0$, and the requirement that at $y, u$, and given $\delta u$, it is invertible with respect to $\delta y$.

# 常微分方程代写

## 数学代写|常微分方程代考Ordinary Differential Equations代写|Existence of optimal controls

$$y^{\prime}=A y+B u, \quad y(a)=y_a$$

$$y(x)=e^{(x-a) A} y_a+e^{x A} \int_a^x e^{-s A} B u(s) d s .$$

$$|\tilde{S}(u)|_{\infty}=\max {t \in[a, b]}|S(u)(t)-S(0)(t)|_2 \leq c \sqrt{(b-a)}|u|{L^2\left((a, b) ; \mathbb{R}^m\right)}$$

## 数学代写|常微分方程代考Ordinary Differential Equations代写|Optimality conditions

$$\min {u \in U{a d}} \hat{J}(u)$$

$$(\nabla \hat{J}(u), v-u) \geq 0, \quad v \in U_{a d} .$$

$$c: H \times U \rightarrow Z, \quad(y, u) \mapsto y^{\prime}-f(\cdot, y, u),$$

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