Piecewise viscosity solutions for stochastic singular control.
Populational adaptive evolution, chemotherapeutic resistance and multiple anti-cancer therapies
Resistance to chemotherapies, particularly to anticancer treatments, is an increasing medical concern. Among the many mechanisms at work in cancers, one of the most important is the selection of tumor cells expressing resistance genes or phenotypes. Motivated by the theory of mutation-selection in adaptive evolution, we propose a model based on a continuous variable that represents the expression level of a resistance gene (or genes, yielding a phenotype) influencing in healthy and tumor cells birth/death...
Problemi di ottimizzazione nella teoria delle code
Prolongement des jeux à deux joueurs de somme nulle. Une théorie abstraite des duels
Propagazione dei massimi e unicità di soluzioni di viscosità di equazioni completamente non lineari del primo e del secondo ordine
Regularity along optimal trajectories of the value function of a Mayer problem
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
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
We formulate an Hamilton-Jacobi partial differential equationon a dimensional manifold , with assumptions of convexity of and regularity of (locally in a neighborhood of in ); we define the “min solution” , a generalized solution; to this end, we view as a symplectic manifold. The definition of “min solution” is suited to proving regularity results about ; in particular, we prove in the first part that the closure of the set where is not regular may be covered by a countable number...
Regularity and variationality of solutions to Hamilton-Jacobi equations. Part I: Regularity (errata)
This errata corrects one error in the 2004 version of this paper [Mennucci, ESAIM: COCV10 (2004) 426–451].
Regularity and variationality of solutions to Hamilton-Jacobi equations. Part I: Regularity
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 solutions of the Hamilton-Jacobi equation
Regularity of convex functions on Heisenberg groups
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
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],...
Relations between extrema of parametrical integrals and ordinary integrals
Relationship between dynamic programming and the maximum principle under state constraints.
Results for an optimal control problem with a semilinear state equation with constrained control
Robust portfolio selection under exponential preferences
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...
Second order conditions for periodic optimal control problems
Second order unbounded parabolic equations in separated form
We prove existence and uniqueness of viscosity solutions of Cauchy problems for fully nonlinear unbounded second order Hamilton-Jacobi-Bellman-Isaacs equations defined on the product of two infinite-dimensional Hilbert spaces H'× H'', where H'' is separable. The equations have a special "separated" form in the sense that the terms involving second derivatives are everywhere defined, continuous and depend only on derivatives with respect to x'' ∈ H'', while the unbounded terms are of first order...
Second-order elliptic integro-differential equations : viscosity solutions' theory revisited