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 $\mathbf{W}^{1,1}_{loc}$ for the map $S_t$, $t>0$ that associates to every given initial data $u_0\in \mathrm{Lip}(\mathbb{R}^N)$ the corresponding solution $S_t u_0$ of a Hamilton-Jacobi equation $$u_t+H\big(x, \nabla_{\!x} u\big)=0\,, \qquad t\geq 0,\quad x\in \mathbb{R}^N,$$ with a convex and coercive Hamiltonian $H=H(x,p)$. We provide upper and lower bounds of order $1/\varepsilon^N$ on the the Kolmogorov $\varepsilon$-entropy in $\mathbf{W}^{1,1}$ of the image through the map $S_t$ 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 $\varepsilon$-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 $L^1$ 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 $u=u(x,t)$ is the state variable and $f_{l}$, $f_{r}$ 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 $u(x,t)\doteq \mathcal{S}_t^{AB} \overline u(x)$ denote the solution of the Cauchy problem for (1), with initial datum $u(\cdot,0)=\overline u$, that satisfy at $x=0$ the interface entropy condition associated to a connection $(A,B)$ (see~\cite{MR2195983}), we analyze the family of profiles that can be attained by (1) at a given time $T>0$: \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 $\mathcal{A}^{AB}(T)$ as a class of functions in $BV_{loc}(\mathbb{R}\setminus\{0\})$ that satisfy suitable Ole\v{\i}nik-type inequalities, and that admit one-sided limits at $x=0$ which satisfy specific conditions related to the interface entropy criterium. Relying on this characterisation, we establish the ${\bf L^1}_{loc}$-compactness of the set of attainable profiles when the initial data $\overline u$ 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