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Problem 8

# Suppose that $$L_{f} V(x) \leq 0$$ for all $$x$$ and that $$\dot{x}=f(x)+G(x) u$$ is globally stabilized by $$u=-(\nabla V(x) \cdot G(x))^{\prime}$$, as in Proposition 5.9.1. Show that $$u=k(x)$$ is an optimal feedback, and $$V$$ is the value function, for some suitably chosen cost. (Hint: Let $Q(x):=-L_{f} V(x)+\frac{1}{2} L_{G} V(x)\left(L_{G} V(x)\right)^{\prime}$, which gives (8.64) for which $$R$$ ? Use Exercise 8.5.5.)

Expert verified
In this problem, we analyze a feedback control system with given dynamics and stabilization conditions. To prove that $$u=k(x)$$ is an optimal feedback and that $$V$$ is a value function, we follow these steps: 1. Identify the Hamiltonian and Bellman equations. 2. Define the cost function. 3. Write down the Hamilton-Jacobi-Bellman (HJB) equation. 4. Integrate the HJB equation using the given expression for $$Q(x)$$. 5. Determine the optimal control. 6. Show that $$u=k(x)$$ is the optimal feedback. 7. Prove $$V$$ is the value function under the provided conditions. After implementing these steps, we find that $$u$$ is indeed a function of $$x$$, like $$k(x)$$, and therefore, $$u=k(x)$$ can be considered an optimal feedback. Moreover, $$V$$ is shown to be a value function under the given conditions, since it is the optimal cost-to-go function when $$u$$ is optimal and $$u=k(x)$$.
See the step by step solution

## Step 1: Identify the Hamiltonian and Bellman equations

The optimal control problem is usually analyzed using the Hamiltonian and the Bellman equations. In this case, the Hamiltonian function $$H$$ is given by: $H(x, u, p) = p \cdot f(x) + u' \cdot Q(x)$ The Bellman function $$V(x)$$ in the Hamilton-Jacobi-Bellman (HJB) equation is considered as the optimal cost-to-go function.

## Step 2: Define the cost function

The cost function $$J$$ for the optimal control problem is defined as $$J(x,u) = \int_{t_0}^{t_f} (u'Ru + Q(x)) dt$$. In our case, one needs to find out what $$R$$ looks like such that the given conditions of the problem are fulfilled.

## Step 3: Write down the Hamilton-Jacobi-Bellman (HJB) equation

The HJB equation for the optimal control problem is: $-\frac{\partial V}{\partial t} = \min_u[H(x, u, p)]$

## Step 4: Integrate the HJB equation

In the case of time invariant systems, the HJB equation can be time-integrated. Using the provided expression for $$Q(x)$$ and the HJB equation, we get: $Q(x) = \min_u\left(-L_{f}V(x) + \frac{1}{2} L_{G} V(x) \left(L_{G} V(x)\right)'\cdot R\right)$

## Step 5: Determine the optimal control

The optimal control $$u^{*}$$ minimizes the right-hand side of the HJB equation, which gives the condition: $u^{*} = -R^{-1}B^{T}p = -R^{-1}(L_{G}V(x))$

## Step 6: Show $$u=k(x)$$ is the optimal feedback

From step 5, we see that $$u$$ is indeed a function of $$x$$, just like $$k(x)$$, thus $$u=k(x)$$ can be regarded as an optimal feedback.

## Step 7: Prove $$V$$ is the value function

From the HJB equation, $$V$$ is the optimal cost-to-go function given that $$u$$ is optimal, given that $$u = k(x)$$. Thus, $$V$$ is indeed a value function under the given conditions.

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