Viscosity Solutions And Optimal Control Problems
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| Author | : Martino Bardi |
| Publisher | : Springer Science & Business Media |
| Total Pages | : 588 |
| Release | : 2009-05-21 |
| Genre | : Science |
| ISBN | : 0817647554 |
This softcover book is a self-contained account of the theory of viscosity solutions for first-order partial differential equations of Hamilton–Jacobi type and its interplay with Bellman’s dynamic programming approach to optimal control and differential games. It will be of interest to scientists involved in the theory of optimal control of deterministic linear and nonlinear systems. The work may be used by graduate students and researchers in control theory both as an introductory textbook and as an up-to-date reference book.
| Author | : Robert James Elliott |
| Publisher | : Harlow, England : Longman Scientific & Technical |
| Total Pages | : 116 |
| Release | : 1987 |
| Genre | : Calculus of variations |
| ISBN | : |
| Author | : Wendell H. Fleming |
| Publisher | : Springer Science & Business Media |
| Total Pages | : 436 |
| Release | : 2006-02-04 |
| Genre | : Mathematics |
| ISBN | : 0387310711 |
This book is an introduction to optimal stochastic control for continuous time Markov processes and the theory of viscosity solutions. It covers dynamic programming for deterministic optimal control problems, as well as to the corresponding theory of viscosity solutions. New chapters in this second edition introduce the role of stochastic optimal control in portfolio optimization and in pricing derivatives in incomplete markets and two-controller, zero-sum differential games.
| Author | : Leonard David Berkovitz |
| Publisher | : CRC Press |
| Total Pages | : 394 |
| Release | : 2012-08-25 |
| Genre | : Mathematics |
| ISBN | : 1466560266 |
Nonlinear Optimal Control Theory presents a deep, wide-ranging introduction to the mathematical theory of the optimal control of processes governed by ordinary differential equations and certain types of differential equations with memory. Many examples illustrate the mathematical issues that need to be addressed when using optimal control techniques in diverse areas. Drawing on classroom-tested material from Purdue University and North Carolina State University, the book gives a unified account of bounded state problems governed by ordinary, integrodifferential, and delay systems. It also discusses Hamilton-Jacobi theory. By providing a sufficient and rigorous treatment of finite dimensional control problems, the book equips readers with the foundation to deal with other types of control problems, such as those governed by stochastic differential equations, partial differential equations, and differential games.
| Author | : Paola Loreti |
| Publisher | : |
| Total Pages | : 44 |
| Release | : 2002 |
| Genre | : |
| ISBN | : |
| Author | : Martino Bardi |
| Publisher | : Springer |
| Total Pages | : 268 |
| Release | : 2006-11-13 |
| Genre | : Mathematics |
| ISBN | : 3540690433 |
The volume comprises five extended surveys on the recent theory of viscosity solutions of fully nonlinear partial differential equations, and some of its most relevant applications to optimal control theory for deterministic and stochastic systems, front propagation, geometric motions and mathematical finance. The volume forms a state-of-the-art reference on the subject of viscosity solutions, and the authors are among the most prominent specialists. Potential readers are researchers in nonlinear PDE's, systems theory, stochastic processes.
| Author | : Maurizio Falcone |
| Publisher | : World Scientific |
| Total Pages | : 256 |
| Release | : 2001 |
| Genre | : Mathematics |
| ISBN | : 9789812799807 |
Geometrical optics and viscosity solutions / A.-P. Blanc, G. T. Kossioris and G. N. Makrakis -- Computation of vorticity evolution for a cylindrical Type-II superconductor subject to parallel and transverse applied magnetic fields / A. Briggs ... [et al.] -- A characterization of the value function for a class of degenerate control problems / F. Camilli -- Some microstructures in three dimensions / M. Chipot and V. Lecuyer -- Convergence of numerical schemes for the approximation of level set solutions to mean curvature flow / K. Deckelnick and G. Dziuk -- Optimal discretization steps in semi-lagrangian approximation of first-order PDEs / M. Falcone, R. Ferretti and T. Manfroni -- Convergence past singularities to the forced mean curvature flow for a modified reaction-diffusion approach / F. Fierro -- The viscosity-duality solutions approach to geometric pptics for the Helmholtz equation / L. Gosse and F. James -- Adaptive grid generation for evolutive Hamilton-Jacobi-Bellman equations / L. Grune -- Solution and application of anisotropic curvature driven evolution of curves (and surfaces) / K. Mikula -- An adaptive scheme on unstructured grids for the shape-from-shading problem / M. Sagona and A. Seghini -- On a posteriori error estimation for constant obstacle problems / A. Veeser.
| Author | : Nizar Touzi |
| Publisher | : Springer Science & Business Media |
| Total Pages | : 219 |
| Release | : 2012-09-25 |
| Genre | : Mathematics |
| ISBN | : 1461442869 |
This book collects some recent developments in stochastic control theory with applications to financial mathematics. We first address standard stochastic control problems from the viewpoint of the recently developed weak dynamic programming principle. A special emphasis is put on the regularity issues and, in particular, on the behavior of the value function near the boundary. We then provide a quick review of the main tools from viscosity solutions which allow to overcome all regularity problems. We next address the class of stochastic target problems which extends in a nontrivial way the standard stochastic control problems. Here the theory of viscosity solutions plays a crucial role in the derivation of the dynamic programming equation as the infinitesimal counterpart of the corresponding geometric dynamic programming equation. The various developments of this theory have been stimulated by applications in finance and by relevant connections with geometric flows. Namely, the second order extension was motivated by illiquidity modeling, and the controlled loss version was introduced following the problem of quantile hedging. The third part specializes to an overview of Backward stochastic differential equations, and their extensions to the quadratic case.
| Author | : P. L. Lions |
| Publisher | : |
| Total Pages | : 37 |
| Release | : 1984 |
| Genre | : |
| ISBN | : |
Recent work by the authors and others has demonstrated the connections between the dynamic programming approach to optimal control theory and to two-person, zero-sum differential games problems and the new notion of viscosity solutions of Hamilton-Jacobi PDE's introduced by M.G. Crandall and P.L. Lions. In particular, it has been proved that the dynamic programming principle implies that the value function is the viscosity solution of the associated Hamilton-Jacobi-Bellman and Isaacs equations. In the present work, it is shown that viscosity super- and subsolutions of these equations must satisfy some inequalities called super- and subdynamic programming principle respectively. This is then used to prove the equivalence between the notion of viscosity solutions and the conditions, introduced by A. Subbotin, concerning the sign of certain generalized directional derivatives. (Author).
| Author | : Nikos Katzourakis |
| Publisher | : Springer |
| Total Pages | : 125 |
| Release | : 2014-11-26 |
| Genre | : Mathematics |
| ISBN | : 3319128299 |
The purpose of this book is to give a quick and elementary, yet rigorous, presentation of the rudiments of the so-called theory of Viscosity Solutions which applies to fully nonlinear 1st and 2nd order Partial Differential Equations (PDE). For such equations, particularly for 2nd order ones, solutions generally are non-smooth and standard approaches in order to define a "weak solution" do not apply: classical, strong almost everywhere, weak, measure-valued and distributional solutions either do not exist or may not even be defined. The main reason for the latter failure is that, the standard idea of using "integration-by-parts" in order to pass derivatives to smooth test functions by duality, is not available for non-divergence structure PDE.
