All Questions
Tagged with oc.optimization-and-control stability
14 questions
4
votes
1
answer
90
views
Do Pareto critical points of a multicriteria optimization problem form an attractor of the dynamical system induced by a descent algorithm?
Let $d\in\mathbb N$, $k\in\mathbb N$ and $f:\mathbb R^d\to\mathbb R^k$ be differentiable. Say that $v\in\mathbb R^d$ is a descent direction at $x\in\mathbb R^d$ if ${\rm D}f(x)v<0$ (component-wise) ...
- 167
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 ...
- 617
3
votes
0
answers
107
views
Stability of a nonlinear dynamical system with non-elementary dynamics
I am trying to prove stability and get a non-asymptotic upper bound on the convergence rate of a nonlinear discrete-time dynamical system, whose dynamics are stated in terms of the (non-elementary) ...
- 31
2
votes
2
answers
1k
views
Routh-Hurwitz criterion for matrices
The Routh-Hurwitz criterion explicitly specifies a finite set of inequalities on the coefficients of a polynomial, necessary and sufficient that all zeros lie in the unit circle or in the left half ...
- 3,873
2
votes
1
answer
102
views
Norm bound in simultaneous stability to semidefinite program
In the context of robust control, I remember hearing that the two following problems are equivalent.
Find $P \succ 0$, such that $A P + P A^{\top} \prec 0$ for all $A \in \mathscr{A}$ where $$\...
- 275
2
votes
0
answers
48
views
Characterization of multifunctions with globally asymptotically stable minimal invariant sets
Let $(X, d)$ be a compact connected metric space. Consider a compact-valued upper semicontinuous multifunction $F: X \rightrightarrows X$. The reachable set $R[x]$ of $F$ from $x\in X$ is defined as:
...
- 111
2
votes
0
answers
61
views
Strong stability of the wave equation with time depending potential
It is well known that the wave equation with frictional damping
$$\eqalign{
& {y_{tt}} = {y_{xx}} - a(t,x){y_t}{\text{ }}{\text{,(t}}{\text{,x)}} \in {\text{ }}(0,\infty ) \times (0,1) \cr
...
- 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 ...
- 617
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(...
- 617
1
vote
1
answer
137
views
Stability of certain second order ODE
I am having a hard time determining the motion of $X$ in the ODE $X''+\nabla f(X)=0$ with initial conditions arbitrary $X(0)$ and zero velocity, i.e. $X'(0) = 0$. $X$ is in $\mathbb{R}^{n}$, and $f$ ...
- 11
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 ...
- 55
1
vote
0
answers
69
views
What is a suitable state feedback adaptive law in discrete time?
I am trying to find a discrete adaptive law and prove stability in an analogous way to a well-known case in continuous time. Let me give a rough explanation of the background.
In continuous time, let ...
- 11
1
vote
0
answers
79
views
Showing a modified system of quadratic equations is stable
I have and $n$ dimensional dynamical system, given by
$\dot{x} = M D(x) P x - \frac{c}{2}x$
$P$ is a full rank $n \times n$ matrix, with $p_{ij} \in [0,c]$, such that $p_{ij}=c-p_{ji}$ for some ...
- 111
0
votes
0
answers
113
views
Reference for matrix Lyapunov function / matrix dynamic system / stability
We usually consider $\dot{x} = f(x)$, where $x$ is a vector.
Now, I want to consider $$\dot{X}=f(X,U),$$ where $X$ is a square matrix $\mathbb{R}^{n\times n}$ state, $U$ is a square matrix variable $\...
- 101