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Problem 3

# Let $$\Sigma_{1}$$ and $$\Sigma_{2}$$ be two systems, and let $$k_{i}: y_{1} \times y_{2} \rightarrow \mathcal{U}_{i}, \quad i=1,2$$ be two maps. We say that the interconnection of $$\Sigma_{1}$$ and $$\Sigma_{2}$$ through $$k_{1}$$ and $$k_{2}$$ is well-posed if for each $\left(x_{1}, x_{2}\right) \in X_{1} \times \mathcal{X}_{2}$ there exist unique functions $$\xi_{i}:[0, \infty) \rightarrow X_{i}, \quad i=1,2$$ such that, with the notations \begin{aligned} \eta_{i}(t) &:=h_{i}\left(\xi_{i}(t)\right) \\ \omega_{i}(t) &:=k_{i}\left(\eta_{1}(t), \eta_{2}(t)\right) \end{aligned} it holds that $$\xi_{i}=\psi_{i}\left(x_{i}, \omega_{i}\right)$$ for $$i=1,2$$.

Expert verified
The well-posedness condition for interconnected systems $$\Sigma_1$$ and $$\Sigma_2$$ holds if there exist unique functions $$\xi_1(t)$$ and $$\xi_2(t)$$ satisfying the following equations: $$\xi_1 = \psi_1\left(x_1, k_1\left(h_1\left(\xi_1(t)\right), h_2\left(\xi_2(t)\right)\right)\right)$$ $$\xi_2 = \psi_2\left(x_2, k_2\left(h_1\left(\xi_1(t)\right), h_2\left(\xi_2(t)\right)\right)\right)$$ These conditions establish the relations between the output $$\eta_i(t)$$, input $$\omega_i(t)$$, and the unique functions $$\xi_i(t)$$ for both systems.
See the step by step solution

## Step 1: Understanding the problem components

First, let's understand the different components of the problem. - $$k_i$$: maps from $$y_1 \times y_2$$ to $$\mathcal{U}_i$$, for both systems $$i=1,2$$. - $$\xi_i$$: a unique function from $$[0, \infty)$$ to $$X_i$$ for both systems $$i=1,2$$. - $$\eta_i(t)$$: represents the output of the system $$i$$ for a specific time $$t$$. - $$\omega_i(t)$$: represents the input to the system $$i$$ at a specific time $$t$$. Our goal is to verify if the well-posedness condition holds true for both systems, given the interconnections through the maps $$k_1$$ and $$k_2$$.

## Step 2: Finding $$\xi_i$$ functions

We need to find the unique functions $$\xi_i:[0, \infty) \longrightarrow X_{i}$$ for $$i=1,2$$ that satisfy the given conditions. Since both systems have similar conditions, we will discuss one system (i=1) and the same procedure can be applied to the other system (i=2). To find the function $$\xi_1$$, we will use the given equation $$\xi_i = \psi_i(x_i, \omega_i)$$. For $$i=1$$, this equation becomes: $$\xi_1 = \psi_1(x_1, \omega_1)$$

## Step 3: Defining $$\omega_i(t)$$

To find the $$\omega_i(t)$$, we are given the following: $$\omega_{i}(t) =k_{i}\left(\eta_{1}(t), \eta_{2}(t)\right)$$ Compute $$\omega_1(t)$$ in terms of given information: $$\omega_1(t) = k_1(\eta_1(t), \eta_2(t))$$

## Step 4: Defining $$\eta_i(t)$$

We are given that $$\eta_i(t)$$ is defined as follows: $$\eta_{i}(t) = h_{i}\left(\xi_{i}(t)\right)$$ Compute $$\eta_1(t)$$ and $$\eta_2(t)$$ using the $$\xi_1(t)$$ function found in Step 2: \begin{aligned} \eta_1(t) &= h_1(\xi_1(t)) \\ \eta_2(t) &= h_2(\xi_2(t)) \end{aligned}

## Step 5: Verifying well-posedness

Since we have already found the functions $$\xi_1(t)$$ and $$\xi_2(t)$$ and the relation between $$\eta_1(t), \eta_2(t), \omega_1(t)$$, and $$\omega_2(t)$$, we now have to check if these conditions are satisfied for both systems. For the first system: $$\xi_1 = \psi_1\left(x_1, k_1\left(h_1\left(\xi_1(t)\right), h_2\left(\xi_2(t)\right)\right)\right)$$ For the second system: $$\xi_2 = \psi_2\left(x_2, k_2\left(h_1\left(\xi_1(t)\right), h_2\left(\xi_2(t)\right)\right)\right)$$ If the conditions above hold, then the interconnection of $$\Sigma_1$$ and $$\Sigma_2$$ through $$k_1$$ and $$k_2$$ is considered well-posed. As the problem statement assumes the existence of unique $$\xi_i$$, this is sufficient for verifying the well-posedness for interconnected systems $$\Sigma_1$$ and $$\Sigma_2$$.

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