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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 ...
Gustave's user avatar
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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 ...
Gustave's user avatar
  • 617
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 ...
Gustave's user avatar
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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(...
Gustave's user avatar
  • 617
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 ...
elmas's user avatar
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