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Problem 2

# Consider the nonlinear, affine in controls, system $$\dot{x}=x^{2}+u$$, with $$\mathcal{U}=[-1,1]$$ (any other compact convex set $$\mathcal{U}$$ could be used). Show that there are states $$x^{0}$$ and times $$T$$ for which $$\mathcal{R}^{T}\left(x^{0}\right)$$ is not compact. Why does this not contradict the discussion in Remark 10.1.10?

Expert verified
The reachable set $$\mathcal{R}^T(x^0)$$ is not compact for the initial state $$x^0 = -1$$ and some time duration $$T$$, as the system will enter an unbounded reachable set when the control input is at its maximum value ($$u=1$$). This does not contradict Remark 10.1.10 because the example given does not satisfy all the conditions mentioned in the remark, which assumes that the reachable set is compact for all $$x^0$$ and $$T$$. The fact that there exists a specific case where the reachable set is not compact shows that the mentioned conditions in Remark 10.1.10 do not hold for this particular example.
See the step by step solution

## Step 1: Analyze the system's dynamics with control input boundaries

We first want to analyze the behavior of the system when the control input $$u$$ reaches its boundaries. Recall that the control input is bounded by the set $$\mathcal{U}=[-1,1]$$, meaning that $$u$$ can take values from the range [-1, 1]. Observe how the system's dynamics will behave at these extreme values: 1. When $$u$$ is at its maximum value, which is 1, the dynamics of the system $$\dot{x} = x^2 + u$$ become $$\dot{x} = x^2 + 1$$. 2. When $$u$$ is at its minimum value, which is -1, the dynamics of the system $$\dot{x} = x^2 + u$$ become $$\dot{x} = x^2 - 1$$.

## Step 2: Choose the initial state $$x^0$$

We will now choose the initial state $$x^0$$ for our analysis. To ensure that the reachable set $$\mathcal{R}^T(x^0)$$ is not compact, we will choose $$x^0 = -1$$, which is also within the control input range. With this choice, we can analyze how the system's dynamics evolve over time, given the different values for the control input $$u$$.

## Step 3: Analyze the system's evolution for different control input values

For the given initial state $$x^0 = -1$$, we will examine how the system evolves over time with the minimum and maximum control input values: 1. For the minimum value of $$u = -1$$, the dynamics of the system are described as: $$\dot{x} = x^2 - 1$$. In this case, $$\dot{x} = (-1)^2 -1 = 0$$. This means that the system will be at a steady-state and will not change over time. 2. For the maximum value of $$u = 1$$, the dynamics of the system are described as: $$\dot{x} = x^2 + 1$$. In this case, $$\dot{x} = (-1)^2 +1 = 2$$, which implies that the system's state is increasing over time. Therefore, with the control input of $$u=1$$, the system will enter an unbounded reachable set. We can conclude that there exists an initial state $$x^0 = -1$$ and a time duration $$T$$ for which the reachable set $$\mathcal{R}^T(x^0)$$ is not compact.

## Step 4: Explain the contradiction with Remark 10.1.10

The discussion in Remark 10.1.10 points out that under certain conditions, the reachable set $$\mathcal{R}^T(x^0)$$ is compact. However, the main condition mentioned in Remark 10.1.10 is that the system should be both nonlinear and affine in controls. In our example, the system is indeed nonlinear and affine in controls. However, the restriction on the control input results in the system entering an unbounded reachable set for the specific initial condition $$x^0 = -1$$. This fact does not contradict Remark 10.1.10, because the given example doesn't satisfy all the conditions mentioned in Remark 10.1.10, besides being nonlinear and affine in controls, Remark 10.1.10 also assumes that the reachable set is compact. Remark 10.1.10 is valid to situations where the reachable set remains compact for all $$x^0$$ and $$T$$.

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