1,721,143 research outputs found
structural Properties of Singularities of Semiconcave Functions
No full textA semiconcave function on an open domain of is a function that can be locally represented as the sum of a concave function plus a smooth one. The local structure of the singular set (non-differentiability points) of such a function is studied in this paper. A new technique is presented to detect singularities that propagate along Lipschitz arcs and, more generally, along sets of higher dimension. This approach is then used to analyze the singular set of the distance function from a closed subset of
Propagation of singularities for solutions of nonlinear first order partial differential equations
No full textFew results are available in the mathematical literature for studying the structure of the singular set of a weak solution u of F (x, u, Du) = 0. This paper provides new techniques to analyse such a set when u is semiconcave and F is a nonlinear convex function with respect to p. The main objective achieved here is a classification of the singularities of u that propagate along Lipschitz arcs. Such a propagation phenomenon is also described by means of a generalized characteristics inclusion
Asymptotic analysis for Hamilton-Jacobi-Bellman equations on Euclidean space
No full textThe long-time average behavior of the value function in the calculus of variations is known to be connected to the existence of the limit of the corresponding Abel means. Still in the Tonelli case, such a limit is in turn related to the existence of solutions of the critical (or ergodic) Hamilton-Jacobi equation. The goal of this paper is to address similar issues when set on the whole Euclidean space and the Hamiltonian fails to be Tonelli. We first study the convergence of the time-averaged value function as the time horizon goes to infinity, proving the existence of the critical constant (Ma\~n\'e critical value) for a general control system. Then, we show that the ergodic equation admits solutions for systems associated with a family of vector fields which satisfies the Lie Algebra rank condition. Finally, we construct a solution to the critical HJB equation on the whole space which coincides with its Lax-Oleinik evolution
Approximate multiplicative controllability for degenerate parabolic problems with Robin boundary conditions
No full textIn this work we study the global approximate multiplicative controllability for a weakly degenerate parabolic Cauchy-Robin problem. The problem is weakly degenerate in the sense that the diffusion coefficient is positive in the interior of the domain and is allowed to vanish at the boundary, provided the reciprocal of the diffusion coefficient is summable. In this paper, we will show that the above system can be steered, in the space of square-summable functions, 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 the initial-state
A stability result for a class of nonlinear integrodifferential equations with kernels
No full textWe study second order nonlinear integro-differential equations in Hilbert spaces with weakly singular convolution kernels obtaining energy estimates
for the solutions. Then, we show that solutions decay exponentially in the energy norm. Finally, we apply these results to
a problem in viscoelasticity
Approximate controllability for linear degenerate parabolic problems with bilinear control
No full textIn this work we study the global approximate multiplicative controllability for the linear degenerate parabolic Cauchy-Neumann problem
\left\{\begin{array}{l}
\displaystyle{v_t-(a(x) v_x)_x =\alpha (t,x)v\,\,\qquad \mbox{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)}~,
\end{array}\right.
with the bilinear control The problem is strongly degenerate in the sense that positive on is allowed to vanish at provided that a certain integrability
condition is fulfilled.
We will show that the above system can be steered in 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
Controllabilità moltiplicativa approssimata di equazioni paraboliche degeneri con applicazioni alla climatologia
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