1,721,034 research outputs found
The concept of value in differential games of survival and viscosity solutions of Hamilton-Jacobi equations
No full textOn Aronsson Equation and Deterministic Optimal Control
No full textWhen Hamiltonians are nonsmooth, we define viscosity solutions of the Aronsson equation and prove that value functions of the corresponding deterministic optimal control problems are solutions if they are bilateral viscosity solutions of the Hamilton-Jacobi-Bellman equation. We characterize such a property in several ways, in particular it follows that a value function which is an absolute minimizer is a bilateral viscosity solution of the HJB equation and these two properties are often equivalent. We also determine that bilateral solutions of HJB equations are unique among absolute minimizers with prescribed boundary conditions
Viscosity and almost everywhere solutions of first-order Carnot-Caratheodory Hamilton-Jacobi equations
No full textWe consider viscosity and distributional derivatives of functions in the directions of a family of vector fields, generators of a Carnot-Carath\`eodory (C-C in brief) metric. In the framework of convex and non coercive Hamilton-Jacobi equations of C-C type we show that viscosity and a.e. subsolutions are equivalent concepts. The latter is a concept related to Lipschitz continuity with respect to the metric generated by the family of vector fields. Under more restrictive assumptions that include Carnot groups, we prove that viscosity solutions of C-C HJ equations are Lipschitz continuous with respect to the corresponding Carnot-Carath\`eodory metric
and satisfy the equation a.e.
Uniqueness results for fully nonlinear degenerate elliptic equations with discontinuous coefficients
No full textIn this paper we prove the comparison principle for viscosity solutions of second order, degenerate elliptic pdes with a discontinuous, inhomogeneous term having discontinuities on Lipschitz surfaces. It is shown that appropriate sub and supersolutions u, v of a Dirichlet type boundary value problem satisfy u <= v in Omega. In particular, continuous viscosity solutions are unique. We also give examples of existence results and apply the comparison principle to prove convergence of approximations
Optimality principles and representation formulas for viscosity solutions of Hamilton-Jacobi equations. I. Equations of unbounded and degenerate control problems without uniqueness
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