Introduction
Of course, the conservation of energy is not violated here. This is a common misconception related to Active Noise Control (ANC). In this 1D problem (I consider this to be a 1D problem since we assume only plane waves and we don’t take into account the angle of incidence at the junction) is shown below why this is the case.
What follows is based on the basic derivations provided in ”Signal Processing for Active Control” by Steven Elliot, which is a highly recommended textbook for those who would like to get started with ANC.
Derivation
I will assume that the reader is familiar with the plane wave equation which is given below
$$ p \left( x, t \right) = A e^{j \left( \omega t - k x \right)} \tag{1} \label{1} $$
where $x$ is the spatial coordinate (a scalar in this 1D case), $t$ is the time dimention, $A$ is the amplitude of the wave, $\omega$ the radial frequency for which $\omega = 2 \pi f$, with $f$ the temporal frequency is true, $k$ is the wavenumber (scalar in this 1D case) for which $k = \frac{\omega}{c} = \frac{2 \pi}{\lambda}$ is true with $c$ the speed of propagation and $\lambda$ the wavelength.
It is easier to drop the time dependence and work in the wavenumber domain, in which equation \eqref{1} is written as
$$ p \left( x \right) = A e^{- j k x} \tag{2} \label{2} $$
Now, I will use a simpler system than the one you provide and then extend to what you have shown in your image. So, below you see an equivalent system (source: ”Signal Processing for Active Control” by Steven Elliot)
In the image, we see three propagating waves where $p_{p+} \left( x \right)$ is the “noise” (I’ll use this term to denote the signal we care to cancel, in the literature this is found as the primary source) propagating on the positive $x$ direction, $p_{s+} \left( x \right)$ and $p_{s-} \left( x \right)$ waves which are the propagating waves in the positive and negative directions respectively due to the “control” source (in the literature this is found as secondary source).
Note that $u$ is the signal of the “control” source and it is in the same frequency $\omega$ (hence same wavenumber $k$) as the “noise” source signal. The waves due to the “control” signal can be written as
$$ \begin{align}
p_{s+} \left( x \right) = B e^{-j k x}, ~~~ x > 0 \tag{3.a} \label{3.a}\\
p_{s-} \left( x \right) = B e^{+j k x}, ~~~ x < 0 \tag{3.b} \label{3.b}
\end{align}$$
where again here $B$ is the amplitude of the waves.
The condition we want to impose is
$$
p_{p+} \left( x \right) + p_{s+} \left( x \right) = 0 \implies
p_{s+} \left( x \right) = -p_{p+} \left( x \right) \implies
B e^{-j k x} = -A e^{-j k x} \implies B = -A \tag{4} \label{4}
$$
By setting the amplitude of the “control” source waves equal to the negative of the “noise” wave we can achieve perfect (in theory only) noise reduction downstream.
Now, what about upstream, towards the direction of the “noise” source? Well, we can calculate the resulting sound field as the summation of the two fields in this region. This is
$$
p_{p+} \left( x \right) + p_{s-} \left( x \right) = A e^{-j k x} - A e^{+j k x} = -A \left( e^{j k x} - e^{-j k x} \right) = -2 j A \sin \left(k x \right) \tag{5} \label{5}
$$
where in the last step the known equality $e^{j x} - e^{-j x} = 2 j \sin \left( x \right)$ has been used. Notice that equation \eqref{5} describes a standing wave, which is the superposition of a positive-going with a negative-going wave. The nodes (pressure nodes in this case) are at $x = 0, \frac{\lambda}{2}, \lambda, \ldots, n \frac{\lambda}{2}, ~~~ n \in \mathbb{N}$ (this is where the function $\sin \left( k x \right)$ is zero). Similarly, at positions $x = \frac{\lambda}{4}, \frac{3 \lambda}{4}, \ldots, \frac{\left(2 n + 1 \right) \lambda}{4}, ~~~ n \in \mathbb{N}$ the amplitude of the resulting wave is $2 A$.
Thus, the energy that was moving towards the right (positive $x$ direction) has now been diverted to the left (negative $x$ direction). This constitutes a realisation of the energy conservation principle.
Extention to your problem
Please note that the above situation is completely equivalent to your system. That is because the phenomena at the junction (you describe it as the ”Summing point”) are exactly the same as those described in the previous section (Derivation) if we assume (lossless) plane wave propagation in an infinite domain. The back-propagating wave from the “control” source you mention is not part of the system and thus it plays absolutely no role in the conservation of energy in the system. The energy in this other system (the propagation of the “back-propagating” wave of the “control” source) is conserved by definition since there are no other sources and the system is lossless by definition.
