The equation provided in the image is:

$\theta_{1}^{T} a(s) P(s) + (\theta_{2}^{T} a(s) + \theta_{0}^{T} a(s)) \lambda_{s} Z(s) = A(s) P(s) - k_{\rho} Z(s) P_{\rho} (s)$

To solve for $\theta_1$, $\theta_2$, and $\theta_0$, we need to separate these terms and express them in terms of other variables. Here's a step-by-step outline:

1. Combine terms involving the thetas on the left-hand side:

   $\theta_{1}^{T} a(s) P(s) + (\theta_{2}^{T} a(s) + \theta_{0}^{T} a(s)) \lambda_{s} Z(s)$

   This simplifies to:

   $\theta_{1}^{T} a(s) P(s) + \theta_{2}^{T} a(s) \lambda_{s} Z(s) + \theta_{0}^{T} a(s) \lambda_{s} Z(s)$

   which can be grouped as:

   $\theta_{1}^{T} a(s) P(s) + (\theta_{2}^{T} + \theta_{0}^{T}) a(s) \lambda_{s} Z(s)$

2. Separate the coefficients of $a(s)$ on the left and the terms on the right:

   Since both sides of the equation must balance, this implies a system of linear equations in terms of $\theta_1$, $\theta_2$, and $\theta_0$.

   Comparing coefficients:

   $\theta_{1}^{T} a(s) P(s) = A(s) P(s)$

   and:

   $(\theta_{2}^{T} + \theta_{0}^{T}) a(s) \lambda_{s} Z(s) = -k_{\rho} Z(s) P_{\rho} (s)$

3. Solve for $\theta_1$:

   From the first equation:

   $\theta_{1}^{T} a(s) = A(s)$

   Therefore:

   $\theta_1 = (a(s))^{-T} A(s) / P(s)$

   if $a(s)$ is invertible.

4. Solve for $\theta_2$ and $\theta_0$:

   From the second equation:

   $(\theta_{2}^{T} + \theta_{0}^{T}) a(s) = -k_{\rho} P_{\rho}(s) / (\lambda_s Z(s))$

   To solve explicitly, we may need further information or additional constraints about $a(s)$, $\lambda_s$, $Z(s)$, and $P_{\rho}(s)$.

Would you like a more detailed solution, or have any specific constraints or values to consider?

Further Questions:

1. Do you have values or expressions for $A(s)$, $P(s)$, and $a(s)$?
2. Can you provide more context or assumptions about the functions involved?
3. Is $a(s)$ guaranteed to be invertible?
4. Are $\theta_1$, $\theta_2$, and $\theta_0$ vectors or scalars?
5. Are there specific boundary conditions or additional equations for these parameters?

Tip:

When solving systems of equations involving parameters like $\theta$, always check for linear independence of the equations to ensure a unique solution.