Studying a nonlinear system under constraints in linear fashion

dotiozeros

Suppose that [itex]a = aBc[/itex] where [itex]a[/itex] and [itex]c[/itex] are vectors and [itex]B[/itex] is some matrix that changes as time "continuously" goes on - making this system dynamical system. But suppose that at any time, if [itex]B[/itex] is linearized into square matrix, it is proven that it is impossible to satisfy [itex]a = aBc[/itex]. Does this mean that in non-linear case, there would not be any case that satisfies [itex]a = aBc[/itex]? Also, if nonlinear case allows satisfying the equation, what would be the condition? If the latter question is somehow ambiguous, answer to the first question is fine.

Caution: [itex]a[/itex] is transposed form of a vector. About [itex]a[/itex] and [itex]c[/itex] - what happens if we separate the case into two - one that has [itex]a[/itex] and [itex]c[/itex] change as time goes on(which means [itex]B[/itex] changes) and one that has these two fixed?