  In this work, we study a new notion involving convergence of microstructures
represented by matrices $B^\epsilon$ related to the classical $H$-convergence
of $A^\epsilon$. It incorporates the interaction between the two
microstructures. This work is about its effects on various aspects : existence,
examples, optimal bounds on emerging macro quantities, application etc. Five
among them are highlighted below : $(1)$ The usual arguments based on
translated inequality, $H$-measures, Compensated Compactness etc for obtaining
optimal bounds are not enough. Additional compactness properties are needed.
$(2)$ Assuming two-phase microstructures, the bounds define naturally four
optimal regions in the phase space of macro quantities. The classically known
single region in the self-interacting case , namely $B^\epsilon= A^\epsilon$
can be recovered from them, a result that indicates we are dealing with a true
extension of the $\mathcal{G}$-closure problem. $(3)$ Optimality of the bounds
is not immediate because of (a priori) non-commutativity of macro-matrices, an
issue not present in the self-interacting case. Somewhat surprisingly though,
commutativity follows a posteriori. $(4)$ From the application to "Optimal
Oscillation-Dissipation Problems", it emerges that oscillations and dissipation
can co-exist optimally and the microstructures behind them need not be the same
though they are closely linked. Furthermore, optimizers are found among
$N$-rank laminates with interfaces. This is a new feature. $(5)$ Explicit
computations in the case of canonical microstructures are performed, in which
we make use of $H$-measure in a novel way.
