Lions, P.L. Optimal control of diffusion processes and HamiltonJacobi -Bellman equations Part I : The dynamic programming principle and applications. Part II: Viscosity solutions and uniqueness. Comm. Partial Diff. Equations 8, , 1101-1174 and 1229-1276., 1983 16  Home/Search   Document Not in Database   Summary   Related Articles   Check

....Viewpoint, Viscosity Solutions The isotope curve diffusions were motivated by recent work [10] which introduces a weak solution for a diffusion related to (11) in arbitrary dimensions. This work was inspired by the earlier works on viscosity solutions of nonlinear partial differential equations [11 13]. The main difficulty of this approach occurs when rI vanishes and (11) becomes undefined. This equation must then be interpreted in some weak sense as in [10] 3.3 Lagrangian Viewpoint, Differential Geometry The Lagrangian approach is based on classical differential geometry mappings or ....

P. L. Lions, "Optimal control of diffusion processes and Hamilton-Jacobi-Bellmam equations I," Comm. Partial Differential Equations, vol. 8, pp. 1101--1134, 1983.

....Such a curvature portrait would be a powerful tool for 3D object recognition. APPENDIX: EXISTENCE AND UNIQUENESS OF SOLUTIONS OF PRINCIPAL CURVATURE FLOW EQUATIONS A theory of viscosity solutions has been developed to study nonlinear second order partial differential equations such as Eq. 2) [19, 20, 21, 22]. The existence of a unique viscosity solution of mean curvature flow equation is proven in Evans [23] and Chen Giga Goto [24] The latter also shows that the same result holds for a more general class of geometric, degenerate parabolic equations. We will apply this result to establish the ....

Lions, P.-L., "Optimal control of diffusion processes and Hamilton-Jacobi-Bellman equations. I", Comm. Partial Diff. Equations, 8, pp.1134-1101, 1983.

.... Sigma ffi j fx 2 Sigma j dist(x; Sigma) ffig : Also, for an open subset O ae Sigma and x 2 O, we use the notation: x;ff O = infft 0 j X(t; x; ff) 2 Og: We present the DPP for (2.1) whose proof will be given in the final section since it is rather complicated. Theorem 2.2. cf. [L]) Assume that (A0 0 ) and (A1 0 ) hold. Let u : Sigma R be a bounded solution of H(x; u; Du) 0 in Sigma: 6 Then, for ffi 0 and x 2 Sigma ffi , u(x) inf ff2A ( Z x;ff Sigma ffi 0 e Gammas f(X(s; x; ff) ff(s) ds e Gamma x;ff Sigma ffi u(X( x;ff ffi ; x; ff) ....

....x 2 2 (0; 1] 1 Gamma e Gammajx 1 j x 2 for x 2 2 ( Gamma1; 0] 1 Gamma (x 2 2)e Gammajx 1 j Gamma1 for x 2 2 ( Gamma2; Gamma1] Notice that the discontinuity of V occurs at (x 1 ; 1) 2 Omega Gamma 11 4 Proof of Theorem 2.2 The basic idea of our proof was obtained by P. L. Lions in [L] for secondorder PDEs. We also refer to [EI] and [BSo] But, in their argument, we need some regularity of solutions. Hence, we will adapt some approximation techniques. Let u be a solution of (2.1) We shall extend u (with the same notation) to the whole space by setting u(x) 1 for x = 2 ....

P.-L. Lions, Optimal control of diffusion processes and Hamilton-Jacobi-Bellman equations: Part 2: Viscosity solutions and uniqueness, Comm. Partial Differential Equation, 8 (1983), 1229-1276.

....set up, it is natural to consider the variational problem in the framework of viscosity solutions, as done by Alvarez [1] for the geometric Brownian motion case. We recall that the notion of viscosity solutions was introduced by Crandal and Lions [9] for first order equations and by Lions [29, 30] for second order equations. The notion of viscosity solutions for integro differential equations was later pursued by Soner [37, 38] and Sayah [34, 35] for certain problems involving a first order local operator, and by Alvarez and Tourin [2] and Pham [32] for problems involving a second order ....

....and Ishii [23] and Katsoulakis [26] for second order equations. In the present paper, we first prove that the value function of our control problem is a constrained viscosity solution of the associated integro differential variational inequality (see Section 4) As observed by Lions (see, e.g. [30]) the general fact that value functions of control problems can be characterized as viscosity solutions of certain partial differential equations is a direct consequence of the dynamic programming principle. For singular control problems, however, the classical approach of Lions fails because the ....

P. L. Lions, Optimal control of diffusion processes and Hamilton-Jacobi-Bellman equations. II. Viscosity solutions and uniqueness, Comm. Partial Differential Equations 8 (1983), no. 11, 1229--1276.

....set up, it is natural to consider the variational problem in the framework of viscosity solutions, as done by Alvarez [1] for the geometric Brownian motion case. We recall that the notion of viscosity solutions was introduced by Crandal and Lions [9] for first order equations and by Lions [29, 30] for second order equations. The notion of viscosity solutions for integro differential equations was later pursued by Soner [37, 38] and Sayah [34, 35] for certain problems involving a first order local operator, and by Alvarez and Tourin [2] and Pham [32] for problems involving a second order ....

P. L. Lions, Optimal control of diffusion processes and Hamilton-Jacobi-Bellman equations. I. The dynamic programming principle and applications, Comm. Partial Differential Equations 8 (1983), no. 10, 1101--1174.

....control problem of (1.6) via Dynamic Programming, and even if the value function is not continuous in general, it satisfies such nonlinear PDE in the viscosity sense in the interior of K, and a generalized Dirichlet boundary condition on K, see P.L. Lions (1983a) 1983b) Ishii (1989) Ishii and Lions (1990). We proved in Bardi and Goatin (1997) that v is indeed the maximal subsolution of a suitable boundary value problem (BV P ) for the HJB equation, see Sect. 2. Thus we can establish if K is invariant by checking whether the constant 1 is the maximal subsolution of (BV P ) and this leads naturally ....

.... : K IR bounded, upper semicontinuous, and subsolution of (BV P )g; and by Z e the following set Z e = fW : O IR is bounded and lower semicontinuous, O is open, K ae O, W is supersolution of F = 0 in O, W 0 on Og: For the definitions of viscosity sub and supersolution we refer to Ishii, Lions (1990) or Crandall, Ishii, Lions (1992) By a subsolution of (BV P ) we mean a subsolution in viscosity sense not only of the PDE in Omega but also of the boundary condition; this means that, for all x 2 K, minfu(x) F (x; u(x) p; Y )g 0 8(p; Y ) 2 J 2; K u(x) 2.6) Invariant sets for ....

Lions, P.L. (1983a). Optimal control of diffusion processes and HamiltonJacobi -Bellman equations. Part 1: The dynamic programming principle and applications, Part 2: Viscosity solutions and uniqueness, Comm. Partial Differential Equations 8, 1101-1174 and 1229-1276.

....not in general a smooth solution of the equation (1.3) especially when the diffusion coefficient is degenerate. One is forced to use a notion of weak solutions such as viscosity solutions introduced by Crandall P.L. Lions (1983 [7] in the deterministic first order case and by P.L. Lions (1983 [19]) in the second order case for diffusion processes. Soner (1986a b [23] 24] has extended the viscosity approach to piecewise deterministic processes with jumps, but restricts to bounded coefficients. Sayah (1991 [22] studied also first order Hamilton Jacobi equations with integral term, under ....

.... Theta IR n ) and v 2 C 2 ( 0; T ] Theta IR n ) In proving the uniqueness result for viscosity solutions of second order equations, it is convenient to give an intrinsic characterization of viscosity solutions. First, let us recall the notion of parabolic semijets as introduced in P.L. Lions [19]. Given v 2 C 0 ( 0; T ] Theta IR n ) and (t; x) 2 [0; T ) Theta IR n , we define the parabolic superjet: P 2; v(t; x) f(p0 ; p; M) 2 IR Theta IR n Theta S n = v(s; y) v(t; x) p0(s Gamma t) p: y Gamma x) 1 2 (y Gamma x) M(y Gamma x) o(js Gamma tj jy Gamma xj ....

[Article contains additional citation context not shown here]

P.L. Lions. Optimal control of diffusion processes and Hamilton-- Jacobi--Bellman equations. Part 1: The dynamic programming principle and applications and Part 2: Viscosity solutions and uniqueness, Comm. P.D.E, 8 (1983), 1101--1174 and 1229--1276.

.... of value functions which in a sense is opposite to that of [7] We start with solutions of the upper and lower Bellman Isaacs equations which exist by the general theory and prove that they must satisfy certain optimality inequalities (see [18] for the deterministic case and also [6] 16] [17] for the case of stochastic control) which in turn yield that solutions are equal to the value functions. These so called suband superoptimality inequalities of dynamic programming are interesting for their own. The proofs presented here use some ideas from [21] and the proof of dynamic ....

P.L. Lions, Optimal control of diffusion processes and Hamilton-Jacobi-Bellman equations, Parts 1,2, Comm. Partial Differential Equations 8 (1983), 1101-1174 and 12291276.

....extend those of our paper [BB1] for the usual Dirichlet problem (see also [BCD] Existence and uniqueness results for viscosity solution of the boundary value problem (0.1) 0.7) were first proved by P. L. Lions for Hamilton Jacobi Bellman equations by means of stochastic control arguments [L1, L2]; also his definition of the relevant part of the boundary was probabilistic. The usual Dirichlet problem for viscosity solutions of (0.1) has a large literature, see [J, CIL, BCESS] and the references therein. The boundary value problem we study here is strongly connected with the Dirichlet ....

Lions, P.L.: Optimal control of diffusion processes and Hamilton-JacobiBellman equations. Part 1: The dynamic programming principle and applications, Part 2: Viscosity solutions and uniqueness, Comm. Partial Differential Equations 8 (1983), 1101-1174 and 1229-1276.

....Tourin and Zariphopoulou (1994) Shreve and Soner (1994) Barles and Soner (1995) and Pichler (1996) The characterization of V as a constrained solution is natural because of the presence of state constraints given by (2. 4) The notion of viscosity solutions was introduced by Crandall and Lions (1983) for first order equations, and by Lions (1983) for second order equations. Constrained viscosity solutions were introduced by Soner (1986) and CapuzzoDolcetta and Lions (1987) for first order equations (see also Ishii and Lions (1990) For a general overview of the theory we refer to the User s ....

....and Soner (1994) Barles and Soner (1995) and Pichler (1996) The characterization of V as a constrained solution is natural because of the presence of state constraints given by (2. 4) The notion of viscosity solutions was introduced by Crandall and Lions (1983) for first order equations, and by Lions (1983) for second order equations. Constrained viscosity solutions were introduced by Soner (1986) and CapuzzoDolcetta and Lions (1987) for first order equations (see also Ishii and Lions (1990) For a general overview of the theory we refer to the User s Guide by Crandall, Ishii and Lions (1994) and ....

[Article contains additional citation context not shown here]

Lions, P.-L. (1983). Optimal control of diffusion processes and HamiltonJacobi -Bellman equations. Part 1: The dynamic programming principle and applications. Part 2: Viscosity solutions and uniqueness. Communications in PDE, 8, pp 1101--1174 and pp 1229--1276.

....uN Ax; DuN GH (x; DuN ; D 2 uN ) 0 in XN : 5.19) Equation (5.19) is the one used in the proof of Theorem 3.8 and it is easy to see that it is the equation in XN corresponding to the control problem with evolution given by (5. 17) Therefore, by the finite dimensional theory (see [24, 35, 38] for results and techniques that adapt to our situation to obtain the dynamic programming principle and Theorem 3.8) the function uN (y 0 ) inf ff2U ad (0; 1;U) IE Z 1 0 e Gammat L(A fi 2 y N (t; y 0 ; ff) ff(t) dt (5.20) is the unique viscosity solution of (5.19) in XN and the ....

P.-L. LIONS, Optimal control of diffusion processes and Hamilton-Jacobi-Bellman equations. Part 1: The Dynamic programming principle and applications. Part 2: Viscosity solutions and uniqueness, Comm. Part. Diff. Eq. 8 (1983) 1101--1174 and 1229--1276.

....[8] A. Bensoussan and J.L Lions[9, 10] where classical PDE approaches are described and to N. El Karoui[12] N.V Krylov[19] and E.D Sontag[25] where these problems are considered from a probabilistic point of view. The more recent approach by viscosity solutions was first introduced in P. L Lions[20, 21, 22] and is presented in the book of W.H Fleming and H.M Soner[13] According to optimal control theory, it is natural to think U as being a solution (in fact, the right solution ) of the Hamilton Jacobi Bellman equation H(x; u; Du;D 2 u) 0 in Omega ; 2) together with the Dirichlet boundary ....

....t; p; M) sup ff2A ae Gamma 1 2 Tr[a(x; ff)M ] Gamma b(x; ff) Delta p t Gamma f(x; ff) oe (4) for any x 2 Omega Gamma t 2 IR, p 2 IR N and M 2 S N (the space of N Theta N symmetric matrices) Here, a(x; ff) oe(x; ff)oe T (x; ff) for any x 2 Omega and ff 2 A (cf. P. L Lions[20, 21, 22] or the book of W.H Fleming and H.M Soner[13] The first standard difficulty which arises in such control problems is that the value function U is not smooth enough to satisfy the above equation in the classical sense; in fact it is only expected to be continuous and may even present ....

[Article contains additional citation context not shown here]

Lions P.L.: Optimal control of diffusion processes and Hamilton-JacobiBellman equations, Part III, in Nonlinear PDE and Appl., S'eminaire du Coll`ege de France, vol V, Pitman (1985).

....[8] A. Bensoussan and J.L Lions[9, 10] where classical PDE approaches are described and to N. El Karoui[12] N.V Krylov[19] and E.D Sontag[25] where these problems are considered from a probabilistic point of view. The more recent approach by viscosity solutions was first introduced in P. L Lions[20, 21, 22] and is presented in the book of W.H Fleming and H.M Soner[13] According to optimal control theory, it is natural to think U as being a solution (in fact, the right solution ) of the Hamilton Jacobi Bellman equation H(x; u; Du;D 2 u) 0 in Omega ; 2) together with the Dirichlet boundary ....

....t; p; M) sup ff2A ae Gamma 1 2 Tr[a(x; ff)M ] Gamma b(x; ff) Delta p t Gamma f(x; ff) oe (4) for any x 2 Omega Gamma t 2 IR, p 2 IR N and M 2 S N (the space of N Theta N symmetric matrices) Here, a(x; ff) oe(x; ff)oe T (x; ff) for any x 2 Omega and ff 2 A (cf. P. L Lions[20, 21, 22] or the book of W.H Fleming and H.M Soner[13] The first standard difficulty which arises in such control problems is that the value function U is not smooth enough to satisfy the above equation in the classical sense; in fact it is only expected to be continuous and may even present ....

[Article contains additional citation context not shown here]

Lions P.L.: Optimal control of diffusion processes and HamiltonJacobi -Bellman equations, Part II: Viscosity solutions and uniqueness, Comm.P.D.E. 8 (1983).

....[8] A. Bensoussan and J.L Lions[9, 10] where classical PDE approaches are described and to N. El Karoui[12] N.V Krylov[19] and E.D Sontag[25] where these problems are considered from a probabilistic point of view. The more recent approach by viscosity solutions was first introduced in P. L Lions[20, 21, 22] and is presented in the book of W.H Fleming and H.M Soner[13] According to optimal control theory, it is natural to think U as being a solution (in fact, the right solution ) of the Hamilton Jacobi Bellman equation H(x; u; Du;D 2 u) 0 in Omega ; 2) together with the Dirichlet boundary ....

....t; p; M) sup ff2A ae Gamma 1 2 Tr[a(x; ff)M ] Gamma b(x; ff) Delta p t Gamma f(x; ff) oe (4) for any x 2 Omega Gamma t 2 IR, p 2 IR N and M 2 S N (the space of N Theta N symmetric matrices) Here, a(x; ff) oe(x; ff)oe T (x; ff) for any x 2 Omega and ff 2 A (cf. P. L Lions[20, 21, 22] or the book of W.H Fleming and H.M Soner[13] The first standard difficulty which arises in such control problems is that the value function U is not smooth enough to satisfy the above equation in the classical sense; in fact it is only expected to be continuous and may even present ....

[Article contains additional citation context not shown here]

Lions P.L.: Optimal control of diffusion processes and Hamilton-JacobiBellman equations, Part I: The dynamic programming principle and applications, Comm. P.D.E. 8 (1983).

No context found.

Lions, P.L. Optimal control of diffusion processes and HamiltonJacobi -Bellman equations Part I : The dynamic programming principle and applications. Part II: Viscosity solutions and uniqueness. Comm. Partial Diff. Equations 8, , 1101-1174 and 1229-1276., 1983 16

No context found.

Lions, P.-L. (1983) "Optimal Control of Diffusion Processes and Hamilton-Jacobi-Bellman Equations", Parts I and II Communications in P.D.E. 8, 1101-1174, 1229-1276.

No context found.

Lions, P.-L. (1983) "Optimal Control of Diffusion Processes and Hamilton-Jacobi-Bellman Equations", Parts I and II Communications in P.D.E. 8, 1101-1174, 1229-1276.

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