All Questions
5 questions
3
votes
2
answers
417
views
Relation between controllability and stability of PDE
In general, when we talk about controllability, we talk about proving the existence of a control input that transfers the state to a desired state at a desired time $T$. However, when we talk about ...
2
votes
0
answers
61
views
Strong stability of the wave equation with time depending potential
It is well known that the wave equation with frictional damping
$$\eqalign{
& {y_{tt}} = {y_{xx}} - a(t,x){y_t}{\text{ }}{\text{,(t}}{\text{,x)}} \in {\text{ }}(0,\infty ) \times (0,1) \cr
...
1
vote
1
answer
174
views
Observability inequality for the 1D transport equation
Let $(a,b) \subset (0,1)$. Consider the following transport equation
$$z_t+z_x=0, \ (t,x)\in(0,T)\times(0,1), \\z(t,0)=0, \ z(0,x)=z_0(x).$$
It is clear that the solution to the above equation is ...
1
vote
1
answer
174
views
Boundary controllability of the heat equation and observation
I'm studying the boundary controllability of the heat equation
\begin{array}{c}
y_{t}=\Delta y\text{ in }\Omega \times (0,T), \\
y=\mathbf{1}_{\Gamma }u\text{ on }\partial \Omega \times (0,T), \\
u(...
1
vote
0
answers
52
views
On Designing Some Optimal Control Problems
In the context of a dynamical systems, some states may not be attainable with scalar controls from $L^1(0,T)$, but they may be reachable with feedback controls.
If we know that the system is null ...