1,721,052 research outputs found
SBV regularity for Hamilton-Jacobi equations with Hamiltonian depending on (t, x)
No full textIn this paper we prove the special bounded variation regularity of the gradient of a viscosity solution of the Hamilton-Jacobi equation ∂tu + H(t, x,Dxu) = 0 in φ ∪ [0, T] × Rn under the hypothesis of uniform convexity of the Hamiltonian H in the last variable. This result extends the result of Bianchini, De Lellis, and Robyr obtained for a Hamiltonian H = H(Dxu) which depends only on the spatial gradient of the solution
SBV-like regularity for Hamilton-Jacobi equations with a convex Hamiltonian
No full textIn this paper we consider a viscosity solution u of the Hamilton-Jacobi equation∂tu+H(Dxu)=0in Ω⊂[0,T]×Rn, where H is smooth and convex. We prove that when d(t, {dot operator}):=H p(D xu(t, {dot operator})), H p:=∇;H is BV for all t∈[0, T] and suitable hypotheses on the Lagrangian L hold, the Radon measure divd(t, {dot operator}) can have Cantor part only for a countable number of t's in [0, T]. This result extends a result of Robyr for genuinely nonlinear scalar balance laws and a result of Bianchini, De Lellis and Robyr for uniformly convex Hamiltonians. © 2012 Elsevier Inc
Pointwise second-order necessary optimality conditions for the Mayer problem with control constraints
No full textThis paper is devoted to second-order necessary optimality conditions for the Mayer optimal control problem when the control set U is a closed subset of m. We show that, in the absence of endpoint constraints, if an optimal control (Formula Presentd.) is singular and integrable, then for almost every t such that (Formula Presentd.) is in the interior of U, both the Goh and a generalized Legendre-Clebsch condition hold true. Moreover, when the control set is a convex polytope, similar conditions are verified on the tangent subspace to U at (Formula Presentd.) for almost all t's such that (Formula Presentd.) lies on the boundary ∂U of U. The same conditions are valid also for U having a smooth boundary at every t where (Formula Presentd.) is singular and locally Lipschitz and (Formula Presentd.). In the presence of a smooth endpoint constraint, these second-order necessary optimality conditions are satisfied whenever the Mayer problem is calm and the maximum principle is abnormal. If it is normal, then analogous results hold true on some smaller subspaces. © 2013 Society for Industrial and Applied Mathematics
A decomposition theorem for BV functions
No full textThe Jordan decomposition states that a function f: R → R is of bounded variation if and only if it can be written as the dierence of two monotone increasing functions. In this paper we generalize this property to real valued BV functions of many variables, extending naturally the concept of monotone function. Our result is an extension of a result obtained by Alberti, Bianchini and Crippa. A counterexample is given which prevents further extensions
Inward pointing trajectories, Normality of the maximum principle and the non occurrence of the Lavrentieff phenomenon in optimal control under state constraints
No full textIt is well known that every strong local minimizer of the Bolza problem under state constraints satisfies a constrained maximum principle. In the absence of constraints qualifications the maximum principle may be abnormal, that is, not involving the cost functions. Normality of the maximum principle can be investigated by studying reachable sets of an associated linear system under linearized state constraints. In this paper we provide sufficient conditions for the existence of solutions to such system and apply them to guarantee the non occurrence of the Lavrentieff phenomenon in optimal control under state constraints. © Heldermann Verlag
Le principali classi di antiossidanti polifenolici naturali nelle sostanze di origine vegetale
No full text
Time-Dependent Focusing Mean-Field Games: The Sub-critical Case
Full text linkWe consider time-dependent viscous mean-field games systems in the case of local, decreasing and unbounded couplings. These systems arise in mean-field game theory, and describe Nash equilibria of games with a large number of agents aiming at aggregation. We prove the existence of weak solutions that are minimizers of an associated non-convex functional, by rephrasing the problem in a convex framework. Under additional assumptions involving the growth at infinity of the coupling, the Hamiltonian, and the space dimension, we show that such minimizers are indeed classical solutions by a blow-up argument and additional Sobolev regularity for the Fokker–Planck equation. We exhibit an example of non-uniqueness of solutions. Finally, by means of a contraction principle, we observe that classical solutions exist just by local regularity of the coupling if the time horizon is short
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