I'm somewhat familiar with the ideas of boundary and internal (source) control for PDE (in particular, for wave equations) but am wondering if the following type of problem is classified/studied; if so, what are some names used to describe the problem; and finally, what papers/references are there?
Suppose $u(t,x)$ is a solution of a wave equation $$ \partial_{tt} u - \nabla \cdot (c^2(x) \nabla u) = 0 \quad (t,x) \in \mathbb R \times \mathbb R^n. $$
Then, if we choose $T>0$ and specify $u_1$, $u_2$ (subject only to some regularity conditions), it is clear that there is a "control" $(f,g)$ so that if we set $(u(0),u_t(0)) = (f,g)$ then $(u(T),u_t(T)) = (u_1,u_2)$.
So that problem is trivial, but what if we place restrictions on the control/states? For example, if we consider only controls of the form $(0,g)$ with $g$ non-negative and compactly supported. Or if we consider only states such that $u_1$ vanishes on some positive-measure set $\omega$. (or both!)
Thanks in advance for you time and consideration.
(in reality, the more general question I've asked above is hopefully in aid of the the specific question below, so any pertinent advice is appreciated)
If $u$ has initial data $(u(0),u_t(0)) = (0,g(x))$, where $g(x) \geq 0$ and is compactly supported in some set $\Omega$, is there a time $T > 0$ and set $\omega$ with positive measure such that $u(T)\vert_{\omega} \equiv 0$ and $u_t(T)\vert_{\omega} \not\equiv 0$? (I hope the answer is "no", by the way)
Some thoughts on this are:
If we choose $v$ to be a solution of the same wave equation with initial values specified at time $t = T$,
$$ v(T) = \chi_\omega u'(T); \quad v'(T) = \chi_\omega \phi $$ where $\phi$ is arbitrary. Then, starting with $$\int\int_0^T v(\partial_{tt}u - \nabla\cdot(c^2(x)\nabla u) \,dt\,dx = 0$$ and integrating by parts, $$ \int u(0) v'(0) + u'(T)v(T)\,dx = \int u(T)v'(T) + u'(0)v(0)\,dx, $$ and plugging-in our assumptions: $$ 0 < \int_\omega (u'(T,x))^2\,dx = \int g(x) v(0,x)\,dx, $$ for every choice of $\phi$; hopefully this leads to a contradiction.
Some cases are easy: e.g. if $\omega$ is outside the domain of influence of $\Omega$ at time $T$, then $\int g(x) v(0,x)\,dx$ is pretty clearly $0$, and we're done. But what about the general case.
I am not much aware of these kind of problems but have come across this book titled Control and nonlinearity which deals with controllability of PDEs and stuff. I hope you will find it useful.