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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 \(i=1,2\).

Short Answer

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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.
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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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