arXiv subjects

Piermarco Cannarsa

Publications and source records attributed to Piermarco Cannarsa.

78 records · Page 5Linked to original sources

arXiv2011

Exterior sphere condition and time optimal control for differential inclusions

The minimum time function $T(\cdot)$ of smooth control systems is known to be locally semiconcave provided Petrov's controllability condition is satisfied. Moreover, such a regularity holds up to the boundary of the target under an inner ball assumption. We generalize this analysis to differential inclusions, replacing the above hypotheses with the continuity of $T(\cdot)$ near the target, and an inner ball property for the multifunction associated with the dynamics. In such a weakened set-up, we prove that the hypograph of $T(\cdot)$ satisfies, locally, an exterior sphere condition. As is well-known, this geometric property ensures most of the regularity results that hold for semiconcave functions, without assuming $T(\cdot)$ to be Lipschitz.

math.OC↗

arXiv2011

Unique continuation and approximate controllability for a degenerate parabolic equation

This paper studies unique continuation for weakly degenerate parabolic equations in one space dimension. A new Carleman estimate of local type is obtained to deduce that all solutions that vanish on the degeneracy set, together with their conormal derivative, are identically equal to zero. An approximate controllability result for weakly degenerate parabolic equations under Dirichlet boundary condition is deduced.

math.AP↗

arXiv2011

Singular gradient flow of the distance function and homotopy equivalence

It is a generally shared opinion that significant information about the topology of a bounded domain $Ω$ of a riemannian manifold $M$ is encoded into the properties of the distance, $d_{\partialΩ}$, %, $d:Ω\rightarrow [0,\infty [$, from the boundary of $Ω$. To confirm such an idea we propose an approach based on the invariance of the singular set of the distance function with respect to the generalized gradient flow of of $d_{\partialΩ}$. As an application, we deduce that such a singular set has the same homotopy type as $Ω$.

math.AP↗

arXiv2011

Approximate controllability for linear degenerate parabolic problems with bilinear control

In this work we study the global approximate multiplicative controllability for the linear degenerate parabolic Cauchy-Neumann problem $$ \{{array}{l} \displaystyle{v_t-(a(x) v_x)_x =α(t,x)v\,\,\qquad {in} \qquad Q_T \,=\,(0,T)\times(-1,1)} [2.5ex] \displaystyle{a(x)v_x(t,x)|_{x=\pm 1} = 0\,\,\qquad\qquad\qquad\,\, t\in (0,T)} [2.5ex] \displaystyle{v(0,x)=v_0 (x) \,\qquad\qquad\qquad\qquad\quad\,\, x\in (-1,1)}, {array}. $$ with the bilinear control $α(t,x)\in L^\infty (Q_T).$ The problem is strongly degenerate in the sense that $a\in C^1([-1,1]),$ positive on $(-1,1),$ is allowed to vanish at $\pm 1$ provided that a certain integrability condition is fulfilled. We will show that the above system can be steered in $L^2(Ω)$ from any nonzero, nonnegative initial state into any neighborhood of any desirable nonnegative target-state by bilinear static controls. Moreover, we extend the above result relaxing the sign constraint on $v_0$.

math.AP↗

arXiv2011

Regularity Results for Eikonal-Type Equations with Nonsmooth Coefficients

Solutions of the Hamilton-Jacobi equation $H(x,-Du(x))=1$, with $H(\cdot,p)$ Hölder continuous and $H(x,\cdot)$ convex and positively homogeneous of degree 1, are shown to be locally semiconcave with a power-like modulus. An essential step of the proof is the ${\mathcal C}^{1,α}$-regularity of the extremal trajectories associated with the multifunction generated by $D_pH$.

math.OC↗

arXiv2009

Hölder estimates in space-time for viscosity solutions of Hamilton-Jacobi equations

It is well-known that solutions to the basic problem in the calculus of variations may fail to be Lipschitz continuous when the Lagrangian depends on t. Similarly, for viscosity solutions to time-dependent Hamilton-Jacobi equations one cannot expect Lipschitz bounds to hold uniformly with respect to the regularity of coefficients. This phenomenon raises the question whether such solutions satisfy uniform estimates in some weaker norm. We will show that this is the case for a suitable Hölder norm, obtaining uniform estimates in (x,t) for solutions to first and second order Hamilton-Jacobi equations. Our results apply to degenerate parabolic equations and require superlinear growth at infinity, in the gradient variables, of the Hamiltonian. Proofs are based on comparison arguments and representation formulas for viscosity solutions, as well as weak reverse Hölder inequalities.

math.OC↗