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Regularity along optimal trajectories of the value function of a Mayer problem

Carlo Sinestrari (2004)

ESAIM: Control, Optimisation and Calculus of Variations

We consider an optimal control problem of Mayer type and prove that, under suitable conditions on the system, the value function is differentiable along optimal trajectories, except possibly at the endpoints. We provide counterexamples to show that this property may fail to hold if some of our conditions are violated. We then apply our regularity result to derive optimality conditions for the trajectories of the system.

Regularity along optimal trajectories of the value function of a Mayer problem

Carlo Sinestrari (2010)

ESAIM: Control, Optimisation and Calculus of Variations

We consider an optimal control problem of Mayer type and prove that, under suitable conditions on the system, the value function is differentiable along optimal trajectories, except possibly at the endpoints. We provide counterexamples to show that this property may fail to hold if some of our conditions are violated. We then apply our regularity result to derive optimality conditions for the trajectories of the system.

Regularity and variationality of solutions to Hamilton-Jacobi equations. Part I : regularity

Andrea C. G. Mennucci (2004)

ESAIM: Control, Optimisation and Calculus of Variations

We formulate an Hamilton-Jacobi partial differential equation H ( x , D u ( x ) ) = 0 on a n dimensional manifold M , with assumptions of convexity of H ( x , · ) and regularity of H (locally in a neighborhood of { H = 0 } in T * M ); we define the “min solution” u , a generalized solution; to this end, we view T * M as a symplectic manifold. The definition of “min solution” is suited to proving regularity results about u ; in particular, we prove in the first part that the closure of the set where u is not regular may be covered by a countable number...

Regularity and variationality of solutions to Hamilton-Jacobi equations. Part I: Regularity

Andrea C.G. Mennucci (2010)

ESAIM: Control, Optimisation and Calculus of Variations

We formulate an Hamilton-Jacobi partial differential equation H( x, D u(x))=0 on a n dimensional manifold M, with assumptions of convexity of H(x, .) and regularity of H (locally in a neighborhood of {H=0} in T*M); we define the “minsol solution” u, a generalized solution; to this end, we view T*M as a symplectic manifold. The definition of “minsol solution” is suited to proving regularity results about u; in particular, we prove in the first part that the closure of the set where...

Regularity of convex functions on Heisenberg groups

Zoltán M. Balogh, Matthieu Rickly (2003)

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

We discuss differentiability properties of convex functions on Heisenberg groups. We show that the notions of horizontal convexity (h-convexity) and viscosity convexity (v-convexity) are equivalent and that h-convex functions are locally Lipschitz continuous. Finally we exhibit Weierstrass-type h-convex functions which are nowhere differentiable in the vertical direction on a dense set or on a Cantor set of vertical lines.

Regularity properties of the distance functions to conjugate and cut loci for viscosity solutions of Hamilton-Jacobi equations and applications in Riemannian geometry

Marco Castelpietra, Ludovic Rifford (2010)

ESAIM: Control, Optimisation and Calculus of Variations

Given a continuous viscosity solution of a Dirichlet-type Hamilton-Jacobi equation, we show that the distance function to the conjugate locus which is associated to this problem is locally semiconcave on its domain. It allows us to provide a simple proof of the fact that the distance function to the cut locus associated to this problem is locally Lipschitz on its domain. This result, which was already an improvement of a previous one by Itoh and Tanaka [Trans. Amer. Math. Soc. 353 (2001) 21–40],...

Robust portfolio selection under exponential preferences

Dariusz Zawisza (2010)

Applicationes Mathematicae

We consider an incomplete market with an untradable stochastic factor and a robust investment problem based on the CARA utility. We formulate it as a stochastic differential game problem, and use Hamilton-Jacobi-Bellman-Isaacs equations to derive an explicit representation of the robust optimal portfolio; the HJBI equation is transformed using a substitution of the Cole-Hopf type. Not only the pure investment problem, but also a problem of robust hedging is taken into account: an agent tries to...

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