147 research outputs found
Flow Stability of Patchy Vector Fields and Robust Feedback Stabilization
The paper is concerned with patchy vector fields, a class of discontinuous,
piecewise smooth vector fields that were introduced in AB to study feedback
stabilization problems. We prove the stability of the corresponding solution
set w.r.t. a wide class of impulsive perturbations. These results yield the
robusteness of patchy feedback controls in the presence of measurement errors
and external disturbances.Comment: 22 page
Compactness estimates for Hamilton-Jacobi equations depending on space
We study quantitative estimates of compactness in
for the map , that associates to every given initial data the corresponding solution of a
Hamilton-Jacobi equation with a convex and coercive Hamiltonian
. We provide upper and lower bounds of order on the
the Kolmogorov -entropy in of the image through
the map of sets of bounded, compactly supported initial data.
Quantitative estimates of compactness, as suggested by P.D. Lax, could provide
a measure of the order of "resolution" and of "complexity" of a numerical
method implemented for this equation. We establish these estimates deriving
accurate a-priori bounds on the Lipschitz, semiconcavity and semiconvexity
constant of a viscosity solution when the initial data is semiconvex. The
derivation of a small time controllability result is also fundamental to
establish the lower bounds on the -entropy.Comment: 36 pages. arXiv admin note: text overlap with arXiv:1403.455
On the optimization of conservation law models at a junction with inflow and flow distribution controls
The paper proposes a general framework to analyze control problems for
conservation law models on a network. Namely we consider a general class of
junction distribution controls and inflow controls and we establish the
compactness in of a class of flux-traces of solutions. We then derive the
existence of solutions for two optimization problems: (I) the maximization of
an integral functional depending on the flux-traces of solutions evaluated at
points of the incoming and outgoing edges; (II) the minimization of the total
variation of the optimal solutions of problem (I). Finally we provide an
equivalent variational formulation of the min-max problem (II) and we discuss
some numerical simulations for a junction with two incoming and two outgoing
edges.Comment: 29 pages, 14 figure
On the construction of nearly time optimal continuous feedback laws around switching manifolds
In this paper, we address the question of the construction of a nearly time optimal feedback law for a minimum time optimal control problem, which is robust with respect to internal and external perturbations. For this purpose we take as starting point an optimal synthesis, which is a suitable collection of optimal trajectories. The construction we exhibit depends exclusively on the initial data obtained from the optimal feedback which is assumed to be known
Attainable profiles for conservation laws with flux function spatially discontinuous at a single point
Consider a scalar conservation law with discontinuous flux
\begin{equation*}\tag{1} \quad u_{t}+f(x,u)_{x}=0, \qquad f(x,u)= \begin{cases}
f_l(u)\ &\text{if}\ x0, \end{cases}
\end{equation*} where is the state variable and , are
strictly convex maps. We study the Cauchy problem for (1) from the point of
view of control theory regarding the initial datum as a control. Letting
denote the solution of the
Cauchy problem for (1), with initial datum , that
satisfy at the interface entropy condition associated to a connection
(see~\cite{MR2195983}), we analyze the family of profiles that can be
attained by (1) at a given time : \begin{equation*}
\mathcal{A}^{AB}(T)=\left\{\mathcal{S}_T^{AB} \,\overline u : \ \overline
u\in{\bf L}^\infty(\mathbb{R})\right\}. \end{equation*} We provide a full
characterization of as a class of functions in
that satisfy suitable Ole\v{\i}nik-type
inequalities, and that admit one-sided limits at which satisfy specific
conditions related to the interface entropy criterium. Relying on this
characterisation, we establish the -compactness of the set of
attainable profiles when the initial data vary in a given class
of uniformly bounded functions, taking values in closed convex sets. We also
discuss some applications of these results to optimization problems arising in
porous media flow models for oil recovery and in traffic flow.Comment: 25 pages, 7 figure
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