It is well-known that solutions to the Hamilton–Jacobi equation $u_t(t,x)+ H(x,ux(t,x)) = 0$ fail to be everywhere differentiable. Nevertheless, suppose a solution $u$ turns out to be dif- ferentiable at a given point $(t , x)$ in the interior of its domain. May then one deduce that $u$ must be continuously differentiable in a neighborhood of $(t, x)$? Although this question has a negative answer in general, our main result shows that it is indeed the case when the proximal subdifferential of $u(t, ·)$ at $x$ is nonempty. Our approach uses the representation of u as the value function of a Bolza problem in the calculus of variations, as well as necessary conditions for such a problem.

Cannarsa, P., Frankowska, H. (2014). From pointwise to local regularity for solutions of Hamilton-Jacobi-Bellman equations. CALCULUS OF VARIATIONS AND PARTIAL DIFFERENTIAL EQUATIONS, 49(3-4), 1061-1074.

From pointwise to local regularity for solutions of Hamilton-Jacobi-Bellman equations

CANNARSA, PIERMARCO;
2014-01-01

Abstract

It is well-known that solutions to the Hamilton–Jacobi equation $u_t(t,x)+ H(x,ux(t,x)) = 0$ fail to be everywhere differentiable. Nevertheless, suppose a solution $u$ turns out to be dif- ferentiable at a given point $(t , x)$ in the interior of its domain. May then one deduce that $u$ must be continuously differentiable in a neighborhood of $(t, x)$? Although this question has a negative answer in general, our main result shows that it is indeed the case when the proximal subdifferential of $u(t, ·)$ at $x$ is nonempty. Our approach uses the representation of u as the value function of a Bolza problem in the calculus of variations, as well as necessary conditions for such a problem.
2014
Pubblicato
Rilevanza internazionale
Articolo
Esperti anonimi
Settore MAT/05 - ANALISI MATEMATICA
English
Con Impact Factor ISI
Hamilton-Jacobi equations; calculus of variations; value function; conjugate point
Cannarsa, P., Frankowska, H. (2014). From pointwise to local regularity for solutions of Hamilton-Jacobi-Bellman equations. CALCULUS OF VARIATIONS AND PARTIAL DIFFERENTIAL EQUATIONS, 49(3-4), 1061-1074.
Cannarsa, P; Frankowska, H
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/2108/115489
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